:: The {J}ordan-H\"older Theorem :: by Marco Riccardi :: :: Received April 20, 2007 :: Copyright (c) 2007-2012 Association of Mizar Users :: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland). :: This code can be distributed under the GNU General Public Licence :: version 3.0 or later, or the Creative Commons Attribution-ShareAlike :: License version 3.0 or later, subject to the binding interpretation :: detailed in file COPYING.interpretation. :: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these :: licenses, or see http://www.gnu.org/licenses/gpl.html and :: http://creativecommons.org/licenses/by-sa/3.0/. environ vocabularies NUMBERS, FUNCT_1, FUNCT_2, SUBSET_1, RELAT_1, TARSKI, ZFMISC_1, SETFAM_1, FINSEQ_1, CARD_3, CARD_1, NAT_1, ARYTM_3, XXREAL_0, XBOOLE_0, ORDINAL4, GROUP_1, STRUCT_0, LATTICES, GROUP_6, ALGSTR_0, BINOP_1, PARTFUN1, GROUP_2, REALSET1, RLSUB_1, PRE_TOPC, GLIB_000, MATRIX_2, QC_LANG1, MSSUBFAM, WELLORD1, EQREL_1, ARYTM_1, NEWTON, INT_1, GROUP_4, NATTRA_1, FINSEQ_2, ISOCAT_1, FINSEQ_3, FINSET_1, ORDINAL2, MEMBERED, ORDINAL1, FINSEQ_5, RFINSEQ, GROUP_9; notations TARSKI, XBOOLE_0, SUBSET_1, XCMPLX_0, ORDINAL1, RELAT_1, FUNCT_1, RELSET_1, FUNCT_2, STRUCT_0, ALGSTR_0, PARTFUN1, FINSEQ_1, ZFMISC_1, CARD_1, XXREAL_2, FINSET_1, INT_1, NAT_1, FINSEQ_2, FINSEQ_3, GROUP_1, GROUP_2, GROUP_3, XXREAL_0, SETFAM_1, GROUP_4, FINSEQ_5, NUMBERS, MEMBERED, MATRIX_2, RFINSEQ, BINOP_1, REALSET1, GROUP_6, NAT_D, RFUNCT_2; constructors BINOP_1, REAL_1, NAT_D, RFINSEQ, BINARITH, FINSEQ_5, REALSET2, GROUP_4, GROUP_6, MATRIX_2, SEQ_1, XXREAL_2, RELSET_1; registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_1, ORDINAL1, FUNCT_2, FINSET_1, NUMBERS, XXREAL_0, XREAL_0, NAT_1, INT_1, MEMBERED, FINSEQ_1, PRE_CIRC, STRUCT_0, GROUP_1, GROUP_2, GROUP_3, GROUP_6, MATRIX_2, VALUED_0, ALGSTR_0, XXREAL_2, CARD_1, RELSET_1; requirements NUMERALS, REAL, SUBSET, BOOLE, ARITHM; definitions GROUP_2, GROUP_6, FINSEQ_3, FINSEQ_5, TARSKI, SUBSET_1, FINSEQ_1, REALSET1, RELAT_1, GROUP_4, ALGSTR_0, FUNCT_2; theorems FINSEQ_1, GROUP_2, GROUP_3, TARSKI, GROUP_6, FINSEQ_2, FUNCT_1, FUNCT_2, RELAT_1, XBOOLE_0, XBOOLE_1, NAT_1, GROUP_1, XREAL_1, RELSET_1, PARTFUN1, FINSEQ_3, INT_1, ZFMISC_1, CARD_1, CARD_2, XCMPLX_1, ORDINAL1, RFUNCT_2, SUBSET_1, MATRIX_2, MATRIX_7, FINSEQ_5, RFINSEQ, XXREAL_0, WSIERP_1, GROUP_4, SETFAM_1, STRUCT_0, NAT_D, XXREAL_2, XREAL_0, XTUPLE_0; schemes XBOOLE_0, FUNCT_1, FINSEQ_1, NAT_1, FUNCT_2, DOMAIN_1; begin :: Actions and Groups with Operators :: ALG I.3.2 Definition 2 definition let O,E be set; let A be Action of O,E; let IT be set; pred IT is_stable_under_the_action_of A means :Def1: for o being Element of O, f being Function of E, E st o in O & f = A.o holds (f .: IT) c= IT; end; Lm1: for O,E being set, A being Action of O,E holds [#]E is_stable_under_the_action_of A proof let O,E be set; let A be Action of O,E; for o being Element of O, f being Function of E, E st o in O & f = A.o holds (f .: [#]E) c= [#]E; hence thesis by Def1; end; definition let O,E be set; let A be Action of O,E; let X be Subset of E; func the_stable_subset_generated_by(X,A) -> Subset of E means :Def2: X c= it & it is_stable_under_the_action_of A & for Y being Subset of E st Y is_stable_under_the_action_of A & X c= Y holds it c= Y; existence proof defpred P[set] means ex B being Subset of E st $1 = B & X c= $1 & B is_stable_under_the_action_of A; consider XX be set such that A1: for Y being set holds Y in XX iff Y in bool E & P[Y] from XBOOLE_0 :sch 1; set M = meet XX; [#]E is_stable_under_the_action_of A by Lm1; then A2: E in XX by A1; then for x being set st x in M holds x in E by SETFAM_1:def 1; then reconsider M as Subset of E by TARSKI:def 3; take M; now let x be set; assume A3: x in X; now let Y be set; assume Y in XX; then ex B being Subset of E st Y = B & X c= Y & B is_stable_under_the_action_of A by A1; hence x in Y by A3; end; hence x in M by A2,SETFAM_1:def 1; end; hence X c= M by TARSKI:def 3; now let o be Element of O; let f be Function of E, E; assume A4: o in O; assume A5: f = A.o; now let y be set; assume A6: y in f .: M; now let Y be set; assume A7: Y in XX; then ex B being Subset of E st Y = B & X c= Y & B is_stable_under_the_action_of A by A1; then A8: (f .: Y) c= Y by A4,A5,Def1; f .: M c= f .: Y by A7,RELAT_1:123,SETFAM_1:3; then f .: M c= Y by A8,XBOOLE_1:1; hence y in Y by A6; end; hence y in M by A2,SETFAM_1:def 1; end; hence (f .: M) c= M by TARSKI:def 3; end; hence M is_stable_under_the_action_of A by Def1; now let Y be Subset of E; assume Y is_stable_under_the_action_of A & X c= Y; then Y in XX by A1; hence M c= Y by SETFAM_1:3; end; hence thesis; end; uniqueness proof let B1,B2 be Subset of E; assume X c= B1 & B1 is_stable_under_the_action_of A & ( for Y being Subset of E st Y is_stable_under_the_action_of A & X c= Y holds B1 c= Y)& X c= B2 &( B2 is_stable_under_the_action_of A & for Y being Subset of E st Y is_stable_under_the_action_of A & X c= Y holds B2 c= Y ); then B1 c= B2 & B2 c= B1; hence thesis by XBOOLE_0:def 10; end; end; definition let O,E be set; let A be Action of O,E; let F be FinSequence of O; func Product(F,A) -> Function of E,E means :Def3: it = id E if len F = 0 otherwise ex PF being FinSequence of Funcs(E,E) st it = PF.(len F) & len PF = len F & PF.1 = A.(F.1) & for n being Nat st n<>0 & n0; defpred P[Element of NAT] means for F being FinSequence of O st len F = $1 & len F <> 0 holds (ex PF being FinSequence of Funcs(E,E), IT being Function of E,E st IT = PF.(len PF) & len PF = len F & PF.1 = A.(F.1) & (for k being Nat st k<>0 & k 0; reconsider G = F | Seg k as FinSequence of O by FINSEQ_1:18; A5: len G = k by A4,FINSEQ_3:53; per cases; suppose A6: len G = 0; set IT=A.(F.1); 1 in Seg len F by A4,A5,A6; then 1 in dom F by FINSEQ_1:def 3; then F.1 in rng F by FUNCT_1:3; then F.1 in O; then F.1 in dom A by FUNCT_2:def 1; then A7: IT in rng A by FUNCT_1:3; set f = the Function of E,E; reconsider IT as Element of Funcs(E,E) by A7; set PF=<*IT*>; ex f being Function st IT = f & dom f = E & rng f c= E by FUNCT_2:def 2; then reconsider IT as Function of E,E by FUNCT_2:2; take PF, IT; len PF = 1 by FINSEQ_1:40; hence IT = PF.(len PF) by FINSEQ_1:40; thus len PF = len F by A4,A5,A6,FINSEQ_1:40; thus PF.1 = A.(F.1) by FINSEQ_1:40; let k be Nat; assume A8: k<>0 & k 0; set g=A.(F.(k+1)); A10: 0+k<=k+1 by XREAL_1:6; A11: 0+10 & k; IT in Funcs(E,E) by FUNCT_2:9; then <*IT*> is FinSequence of Funcs(E,E) by FINSEQ_1:74; then reconsider PF as FinSequence of Funcs(E,E) by FINSEQ_1:75; take PF, IT; A18: len PF = len G + len <*IT*> by A14,FINSEQ_1:22 .= k + 1 by A5,FINSEQ_1:39; then len PF = len PFk + 1 by A4,A14,FINSEQ_3:53; hence A19: IT=PF.(len PF) & len PF=len F by A4,A18,FINSEQ_1:42; 0+10; assume n= k; then A.(F.(n+1)) = g by A21,XXREAL_0:1; then reconsider g9=A.(F.(n+1)) as Function of E,E; A23: n = k by A21,A22,XXREAL_0:1; then reconsider f9=PF.n as Function of E,E by A17,FINSEQ_1:def 7; take f9,g9; thus f9 = PF.n & g9 = A.(F.(n+1)); thus thesis by A17,A18,A19,A23,FINSEQ_1:def 7; end; suppose A24: n < k; A25: 0+10 & k0 & k0 & k0; then A44: 0+1 < k+1 by XREAL_1:6; (ex f1,g1 be Function of E,E st f1=PF1.k & g1=A.(F.(k+1)) & PF1.(k+1)=f1* g1 )& ex f2,g2 be Function of E,E st f2=PF2.k & g2=A.(F.(k+1)) & PF2.(k+1) =f2*g2 by A33,A35,A38,A42,A43; hence PF1.(k+1) = PF2.(k+1) by A40,A41,A44,NAT_1:13; end; end; hence thesis; end; A45: P[0]; for k be Nat holds P[k] from NAT_1:sch 2(A45,A39); hence IT1=IT2 by A32,A33,A36,FINSEQ_1:14; end; hence thesis; end; consistency; end; :: ALG I.3.4 Definition 6 definition let O be set; let G be Group; let IT be Action of O, the carrier of G; attr IT is distributive means :Def4: for o being Element of O st o in O holds IT.o is Homomorphism of G, G; end; definition let O be set; struct (multMagma) HGrWOpStr over O (# carrier -> set, multF -> BinOp of the carrier, action -> Action of O, the carrier #); end; registration let O be set; cluster non empty for HGrWOpStr over O; existence proof set A = the non empty set,m = the BinOp of A,h = the Action of O,A; take HGrWOpStr(#A,m,h#); thus thesis; end; end; definition let O be set; let IT be non empty HGrWOpStr over O; attr IT is distributive means :Def5: for G being Group, a being Action of O, the carrier of G st a = the action of IT & the multMagma of G = the multMagma of IT holds a is distributive; end; Lm2: for O,E being set holds [:O,{id E}:] is Action of O, E proof let O,E be set; set h = [:O,{id E}:]; now let x be set; assume x in {id E}; then reconsider f=x as Function of E,E by TARSKI:def 1; f in Funcs(E,E) by FUNCT_2:9; hence x in Funcs(E,E); end; then {id E} c= Funcs(E,E) by TARSKI:def 3; then reconsider h as Relation of O,Funcs(E,E) by ZFMISC_1:95; A1: now thus (Funcs(E,E)={} implies O={}) implies O = dom h proof assume Funcs(E,E)={} implies O={}; now let x be set; assume A2: x in O; consider y be set such that A3: y=id E; take y; y in {id E} by A3,TARSKI:def 1; hence [x,y] in h by A2,ZFMISC_1:def 2; end; hence thesis by RELSET_1:9; end; assume O = {}; hence h = {}; end; now let x,y1,y2 be set; assume that A4: [x,y1] in h and A5: [x,y2] in h; consider x9,y9 be set such that x9 in O and A6: y9 in {id E} & [x,y1]=[x9,y9] by A4,ZFMISC_1:def 2; A7: y9=id E & y1=y9 by A6,TARSKI:def 1,XTUPLE_0:1; consider x99,y99 be set such that x99 in O and A8: y99 in {id E} and A9: [x,y2]=[x99,y99] by A5,ZFMISC_1:def 2; y99=id E by A8,TARSKI:def 1; hence y1 = y2 by A9,A7,XTUPLE_0:1; end; then reconsider h as PartFunc of O,Funcs(E,E) by FUNCT_1:def 1; h is Action of O, E by A1,FUNCT_2:def 1; hence thesis; end; Lm3: for O being set, G being strict Group holds ex H being non empty HGrWOpStr over O st H is strict distributive Group-like associative & G = the multMagma of H proof let O be set; let G be strict Group; reconsider h=[:O,{id the carrier of G}:] as Action of O, the carrier of G by Lm2; set A = the carrier of G; set m = the multF of G; set GO = HGrWOpStr(#A,m,h#); reconsider GO as non empty HGrWOpStr over O; reconsider G9=GO as non empty multMagma; A1: now set e=1_G; reconsider e9=e as Element of G9; take e9; let h9 be Element of G9; reconsider h=h9 as Element of G; set g=h"; reconsider g9=g as Element of G9; h9*e9 = h*e .= h by GROUP_1:def 4; hence h9 * e9 = h9; e9*h9 = e*h .= h by GROUP_1:def 4; hence e9 * h9 = h9; take g9; h9*g9 = h*g .= 1_G by GROUP_1:def 5; hence h9 * g9 = e9; g9*h9 = g*h .= 1_G by GROUP_1:def 5; hence g9 * h9 = e9; end; take GO; A2: now let G99 be Group; let a be Action of O, the carrier of G99; assume A3: a = the action of GO; assume A4: the multMagma of G99 = the multMagma of GO; now let o be Element of O; assume o in O; then o in dom h by FUNCT_2:def 1; then [o,h.o] in [:O,{id the carrier of G99}:] by A4,FUNCT_1:1; then consider x,y be set such that x in O and A5: y in {id the carrier of G99} & [o,h.o]=[x,y] by ZFMISC_1:def 2; y = id the carrier of G99 & h.o = y by A5,TARSKI:def 1,XTUPLE_0:1; hence a.o is Homomorphism of G99, G99 by A3,GROUP_6:38; end; hence a is distributive by Def4; end; now let x9,y9,z9 be Element of G9; reconsider x=x9,y=y9,z=z9 as Element of G; (x9*y9)*z9 = (x*y)*z .= x*(y*z) by GROUP_1:def 3; hence (x9*y9)*z9 = x9*(y9*z9); end; hence thesis by A1,A2,Def5,GROUP_1:def 2,def 3; end; registration let O be set; cluster strict distributive Group-like associative for non empty HGrWOpStr over O; existence proof set G = the strict Group; consider H be non empty HGrWOpStr over O such that A1: H is strict distributive Group-like associative and the multMagma of H = G by Lm3; take H; thus thesis by A1; end; end; :: ALG I.4.2 Definition 2 definition let O be set; mode GroupWithOperators of O is distributive Group-like associative non empty HGrWOpStr over O; end; definition let O be set; let G be GroupWithOperators of O; let o be Element of O; func G^o -> Homomorphism of G, G equals :Def6: (the action of G).o if o in O otherwise id the carrier of G; correctness proof now assume A1: o in O; consider G9 be Group such that A2: the multMagma of G9 = the multMagma of G; reconsider a=the action of G as Action of O, the carrier of G9 by A2; a is distributive by A2,Def5; then reconsider f9=a.o as Homomorphism of G9, G9 by A1,Def4; reconsider f=f9 as Function of G, G by A2; now let g1, g2 be Element of G; reconsider g19=g1,g29=g2 as Element of G9 by A2; f.(g1 * g2) = f9.(g19 * g29) by A2 .= f9.g19 * f9.g29 by GROUP_6:def 6 .= (the multF of G).(f.g1,f.g2) by A2; hence f.(g1 * g2) = f.g1 * f.g2; end; hence (the action of G).o is Homomorphism of G,G by GROUP_6:def 6; end; hence thesis by GROUP_6:38; end; end; definition let O be set; let G be GroupWithOperators of O; mode StableSubgroup of G -> distributive Group-like associative non empty HGrWOpStr over O means :Def7: it is Subgroup of G & for o being Element of O holds it^o = (G^o)|the carrier of it; correctness proof set H=G; take H; thus thesis by GROUP_2:54; end; end; Lm4: for O being set, G being GroupWithOperators of O holds the HGrWOpStr of G is StableSubgroup of G proof let O be set; let G be GroupWithOperators of O; reconsider G9 = the HGrWOpStr of G as non empty multMagma; A1: now set e=1_G; reconsider e9=e as Element of G9; take e9; let h9 be Element of G9; reconsider h=h9 as Element of G; set g=h"; reconsider g9=g as Element of G9; h9*e9 = h*e .= h by GROUP_1:def 4; hence h9 * e9 = h9; e9*h9 = e*h .= h by GROUP_1:def 4; hence e9 * h9 = h9; take g9; h9*g9 = h*g .= 1_G by GROUP_1:def 5; hence h9 * g9 = e9; g9*h9 = g*h .= 1_G by GROUP_1:def 5; hence g9 * h9 = e9; end; now let x9,y9,z9 be Element of G9; reconsider x=x9,y=y9,z=z9 as Element of G; (x9*y9)*z9 = (x*y)*z .= x*(y*z) by GROUP_1:def 3; hence (x9*y9)*z9 = x9*(y9*z9); end; then reconsider G9 as strict Group-like associative non empty HGrWOpStr over O by A1,GROUP_1:def 2,def 3; for G being Group, a being Action of O, the carrier of G st a = the action of G9 & the multMagma of G = the multMagma of G9 holds a is distributive by Def5; then reconsider G9 as distributive Group-like associative non empty HGrWOpStr over O by Def5; A2: now let o be Element of O; A3: now per cases; suppose A4: o in O; then G9^o=(the action of G9).o by Def6; hence G9^o = G^o by A4,Def6; end; suppose A5: not o in O; then G9^o = id the carrier of G9 by Def6; hence G9^o = G^o by A5,Def6; end; end; dom(G9^o) = the carrier of G by FUNCT_2:def 1; hence G9^o = (G^o)|the carrier of G9 by A3,RELAT_1:69; end; dom(the multF of G) = [:the carrier of G9,the carrier of G9:] by FUNCT_2:def 1; then the multF of G9 = (the multF of G)||the carrier of G9 by RELAT_1:68; then G9 is Subgroup of G by GROUP_2:def 5; hence thesis by A2,Def7; end; registration let O be set; let G be GroupWithOperators of O; cluster strict for StableSubgroup of G; correctness proof reconsider G9 = the HGrWOpStr of G as StableSubgroup of G by Lm4; take G9; thus thesis; end; end; :: like GROUP_2:68 Lm5: for O being set, G being GroupWithOperators of O, H1,H2 being strict StableSubgroup of G st the carrier of H1 = the carrier of H2 holds H1=H2 proof let O be set; let G be GroupWithOperators of O; let H1,H2 be strict StableSubgroup of G; reconsider H19=H1,H29=H2 as Subgroup of G by Def7; A1: dom the action of H2 = O by FUNCT_2:def 1 .= dom the action of H1 by FUNCT_2:def 1; assume A2: the carrier of H1 = the carrier of H2; A3: now let x be set; assume A4: x in dom the action of H2; then reconsider o=x as Element of O; A5: H1^o = (the action of H1).o by A4,Def6; H1^o = (G^o)|the carrier of H2 by A2,Def7 .= H2^o by Def7; hence (the action of H1).x = (the action of H2).x by A4,A5,Def6; end; the multMagma of H19 = the multMagma of H29 by A2,GROUP_2:59; hence thesis by A1,A3,FUNCT_1:2; end; definition let O be set; let G be GroupWithOperators of O; func (1).G -> strict StableSubgroup of G means :Def8: the carrier of it = { 1_G}; existence proof set G9=(1).G; consider H be non empty HGrWOpStr over O such that A1: H is strict distributive Group-like associative and A2: G9 = the multMagma of H by Lm3; reconsider H as strict GroupWithOperators of O by A1; A3: the carrier of H c= the carrier of G by A2,GROUP_2:def 5; the multF of H = (the multF of G)||the carrier of H by A2,GROUP_2:def 5; then A4: H is Subgroup of G by A3,GROUP_2:def 5; now let o be Element of O; reconsider f9=H^o,f=(G^o)|the carrier of H as Function; A5: dom f = dom((G^o)*(id the carrier of H)) by RELAT_1:65 .= dom(G^o) /\ the carrier of H by FUNCT_1:19 .= (the carrier of G) /\ the carrier of H by FUNCT_2:def 1 .= the carrier of (1).G by A2,A3,XBOOLE_1:28; A6: now let x be set; assume A7: x in dom f; then A8: x in dom id the carrier of H by A2,A5; x in {1_G} by A5,A7,GROUP_2:def 7; then A9: x = 1_G by TARSKI:def 1; then x = 1_H by A4,GROUP_2:44; then A10: f9.x = 1_H by GROUP_6:31; f.x = ((G^o)*(id the carrier of H)).x by RELAT_1:65 .= (G^o).((id the carrier of H).x) by A8,FUNCT_1:13 .= (G^o).x by A2,A5,A7,FUNCT_1:18 .= 1_G by A9,GROUP_6:31; hence f.x = f9.x by A4,A10,GROUP_2:44; end; dom f9= the carrier of (1).G by A2,FUNCT_2:def 1; hence H^o = (G^o)|the carrier of H by A5,A6,FUNCT_1:2; end; then reconsider H as strict StableSubgroup of G by A4,Def7; take H; thus thesis by A2,GROUP_2:def 7; end; uniqueness by Lm5; end; definition let O be set; let G be GroupWithOperators of O; func (Omega).G -> strict StableSubgroup of G equals the HGrWOpStr of G; correctness by Lm4; end; definition let O be set; let G be GroupWithOperators of O; let IT be StableSubgroup of G; attr IT is normal means :Def10: for H being strict Subgroup of G st H = the multMagma of IT holds H is normal; end; registration let O be set; let G be GroupWithOperators of O; cluster strict normal for StableSubgroup of G; existence proof set H=(1).G; set H9=H; reconsider H as StableSubgroup of G; take H9; now reconsider G9=G as Group; let H99 be strict Subgroup of G; assume A1: H99 = the multMagma of H; A2: the multF of (1).G9 = (the multF of G9)||the carrier of (1).G9 by GROUP_2:def 5; the carrier of (1).G9 = {1_G9} by GROUP_2:def 7 .= the carrier of (1).G by Def8; hence H99 is normal by A1,A2,GROUP_2:def 5; end; hence thesis by Def10; end; end; registration let O be set; let G be GroupWithOperators of O; let H be StableSubgroup of G; cluster normal for StableSubgroup of H; existence proof reconsider H9=(1).H as GroupWithOperators of O; reconsider H9 as StableSubgroup of H; take H9; now let H99 be strict Subgroup of H; reconsider H as Group; assume the multMagma of H9 = H99; then the carrier of H99 = {1_H} by Def8; then H99 = (1).H by GROUP_2:def 7; hence H99 is normal; end; hence thesis by Def10; end; end; registration let O be set; let G be GroupWithOperators of O; cluster (1).G -> normal; correctness proof now reconsider G9=G as Group; let H be strict Subgroup of G; reconsider H9=H as strict Subgroup of G9; assume H = the multMagma of (1).G; then the carrier of H = {1_G} by Def8; then H9 = (1).G9 by GROUP_2:def 7; hence H is normal; end; hence thesis by Def10; end; cluster (Omega).G -> normal; correctness proof now reconsider G9=G as Group; let H be strict Subgroup of G; reconsider H9=H as strict Subgroup of G9; assume H = the multMagma of (Omega).G; then H9 = (Omega).G9; hence H is normal; end; hence thesis by Def10; end; end; definition let O be set; let G be GroupWithOperators of O; func the_stable_subgroups_of G -> set means :Def11: for x being set holds x in it iff x is strict StableSubgroup of G; existence proof defpred P[set,set] means ex H being strict StableSubgroup of G st $2 = H & $1 = the carrier of H; defpred P[set] means ex H being strict StableSubgroup of G st $1 = the carrier of H; consider B being set such that A1: for x being set holds x in B iff x in bool the carrier of G & P[x] from XBOOLE_0:sch 1; A2: for x,y1,y2 being set st P[x,y1] & P[x,y2] holds y1 = y2 by Lm5; consider f being Function such that A3: for x,y being set holds [x,y] in f iff x in B & P[x,y] from FUNCT_1:sch 1(A2); for x being set holds x in B iff ex y being set st [x,y] in f proof let x be set; thus x in B implies ex y being set st [x,y] in f proof assume A4: x in B; then consider H being strict StableSubgroup of G such that A5: x = the carrier of H by A1; reconsider y = H as set; take y; thus thesis by A3,A4,A5; end; given y be set such that A6: [x,y] in f; thus thesis by A3,A6; end; then A7: B = dom f by XTUPLE_0:def 12; for y being set holds y in rng f iff y is strict StableSubgroup of G proof let y be set; thus y in rng f implies y is strict StableSubgroup of G proof assume y in rng f; then consider x be set such that A8: x in dom f & y = f.x by FUNCT_1:def 3; [x,y] in f by A8,FUNCT_1:def 2; then ex H being strict StableSubgroup of G st y = H & x = the carrier of H by A3; hence thesis; end; assume y is strict StableSubgroup of G; then reconsider H = y as strict StableSubgroup of G; reconsider x = the carrier of H as set; H is Subgroup of G by Def7; then the carrier of H c= the carrier of G by GROUP_2:def 5; then A9: x in dom f by A1,A7; then [x,y] in f by A3,A7; then y = f.x by A9,FUNCT_1:def 2; hence thesis by A9,FUNCT_1:def 3; end; hence thesis; end; uniqueness proof defpred P[set] means $1 is strict StableSubgroup of G; let A1,A2 be set; assume A10: for x being set holds x in A1 iff P[x]; assume A11: for x being set holds x in A2 iff P[x]; thus thesis from XBOOLE_0:sch 2(A10,A11); end; end; registration let O be set; let G be GroupWithOperators of O; cluster the_stable_subgroups_of G -> non empty; correctness proof (1).G in the_stable_subgroups_of G by Def11; hence thesis; end; end; definition let IT be Group; attr IT is simple means :Def12: IT is not trivial & not ex H being strict normal Subgroup of IT st H <> (Omega).IT & H <> (1).IT; end; Lm6: Group_of_Perm 2 is simple proof set G = Group_of_Perm 2; A1: now let H be strict normal Subgroup of G; assume A2: H <> (Omega).G; assume A3: H <> (1).G; 1_G in H by GROUP_2:46; then 1_G in the carrier of H by STRUCT_0:def 5; then {1_G} c= the carrier of H by ZFMISC_1:31; then {<*1,2*>} c= the carrier of H by FINSEQ_2:52,MATRIX_2:24; then A4: <*1,2*> in the carrier of H by ZFMISC_1:31; the carrier of H c= the carrier of G by GROUP_2:def 5; then A5: the carrier of H c= {<*1,2*>,<*2,1*>} by MATRIX_2:def 10,MATRIX_7:3; per cases by A5,ZFMISC_1:36; suppose the carrier of H = {}; hence contradiction; end; suppose the carrier of H = {<*1,2*>}; then {1_G} = the carrier of H by FINSEQ_2:52,MATRIX_2:24; hence contradiction by A3,GROUP_2:def 7; end; suppose the carrier of H = {<*2,1*>}; then <*2,1*>.1 = <*1,2*>.1 by A4,TARSKI:def 1; then 2 = <*1,2*>.1 by FINSEQ_1:44; hence contradiction by FINSEQ_1:44; end; suppose the carrier of H = {<*1,2*>,<*2,1*>}; then the carrier of H = the carrier of G by MATRIX_2:def 10,MATRIX_7:3; hence contradiction by A2,GROUP_2:61; end; end; now assume G is trivial; then consider e be set such that A6: the carrier of G = {e} by GROUP_6:def 2; Permutations 2 = {e} by A6,MATRIX_2:def 10; then <*2,1*> = <*1,2*> by MATRIX_7:3,ZFMISC_1:5; then 2 = <*1,2*>.1 by FINSEQ_1:44; hence contradiction by FINSEQ_1:44; end; hence thesis by A1,Def12; end; registration cluster strict simple for Group; existence by Lm6; end; :: ALG I.4.4 Definition 7 definition let O be set; let IT be GroupWithOperators of O; attr IT is simple means :Def13: IT is not trivial & not ex H being strict normal StableSubgroup of IT st H <> (Omega).IT & H <> (1).IT; end; Lm7: for O being set, G being GroupWithOperators of O, N being normal StableSubgroup of G holds the multMagma of N is strict normal Subgroup of G proof let O be set; let G be GroupWithOperators of O; let N be normal StableSubgroup of G; set H = the multMagma of N; reconsider H as non empty multMagma; now set e=1_N; reconsider e9=e as Element of H; take e9; let h9 be Element of H; reconsider h=h9 as Element of N; set g=h"; reconsider g9=g as Element of H; h9*e9 = h*e .= h by GROUP_1:def 4; hence h9 * e9 = h9; e9*h9 = e*h .= h by GROUP_1:def 4; hence e9 * h9 = h9; take g9; h9*g9 = h*g .= 1_N by GROUP_1:def 5; hence h9 * g9 = e9; g9*h9 = g*h .= 1_N by GROUP_1:def 5; hence g9 * h9 = e9; end; then reconsider H as Group-like non empty multMagma by GROUP_1:def 2; N is Subgroup of G by Def7; then the carrier of H c= the carrier of G & the multF of H = (the multF of G) || the carrier of H by GROUP_2:def 5; then reconsider H as Subgroup of G by GROUP_2:def 5; H is normal by Def10; hence thesis; end; Lm8: for G1, G2 being Group, A1 being Subset of G1, A2 being Subset of G2, H1 being strict Subgroup of G1, H2 being strict Subgroup of G2 st the multMagma of G1 = the multMagma of G2 & A1 = A2 & H1 = H2 holds A1 * H1 = A2 * H2 & H1 * A1 = H2 * A2 proof let G1, G2 be Group; let A1 be Subset of G1; let A2 be Subset of G2; let H1 be strict Subgroup of G1; let H2 be strict Subgroup of G2; assume A1: the multMagma of G1 = the multMagma of G2; A2: now let A1,B1 be Subset of G1; let A2,B2 be Subset of G2; set X={g*h where g,h is Element of G1: g in A1 & h in B1}; set Y={g*h where g,h is Element of G2: g in A2 & h in B2}; assume A3: A1=A2 & B1=B2; A4: now let x be set; assume x in X; then consider g,h be Element of G1 such that A5: x=g*h & g in A1 & h in B1; set h9=h; set g9=g; reconsider g9,h9 as Element of G2 by A1; g*h = g9*h9 by A1; hence x in Y by A3,A5; end; now let x be set; assume x in Y; then consider g,h be Element of G2 such that A6: x=g*h & g in A2 & h in B2; reconsider g9=g,h9=h as Element of G1 by A1; g*h = g9*h9 by A1; hence x in X by A3,A6; end; hence X=Y by A4,TARSKI:1; end; assume A7: A1 = A2; assume A8: H1 = H2; hence A1 * H1 = A2 * H2 by A7,A2; thus thesis by A7,A8,A2; end; registration let O be set; cluster strict simple for GroupWithOperators of O; existence proof set Gp2 = Group_of_Perm 2; consider G be non empty HGrWOpStr over O such that A1: G is strict distributive Group-like associative and A2: Gp2 = the multMagma of G by Lm3; reconsider G as strict GroupWithOperators of O by A1; take G; now assume A3: G is not simple; per cases by A3,Def13; suppose G is trivial; hence contradiction by A2,Def12,Lm6; end; suppose A4: ex H being strict normal StableSubgroup of G st H <> (Omega).G & H <> (1).G; reconsider G9 = G as Group; consider H be strict normal StableSubgroup of G such that A5: H <> (Omega).G and A6: H <> (1).G by A4; reconsider H9 = the multMagma of H as strict normal Subgroup of G by Lm7; reconsider H9 as strict normal Subgroup of G9; set H99=H9; the carrier of H99 c= the carrier of G9 & the multF of H99 = (the multF of G9)||the carrier of H99 by GROUP_2:def 5; then reconsider H99 as strict Subgroup of Gp2 by A2,GROUP_2:def 5; now let A be Subset of Gp2; reconsider A9=A as Subset of G9 by A2; A * H99 = A9 * H9 by A2,Lm8 .= H9 * A9 by GROUP_3:120; hence A * H99 = H99 * A by A2,Lm8; end; then reconsider H99 as strict normal Subgroup of Gp2 by GROUP_3:120; A7: now reconsider e = 1_Gp2 as Element of G by A2; A8: now let h be Element of G; reconsider h9=h as Element of Gp2 by A2; h * e = h9 * 1_Gp2 by A2 .= h9 by GROUP_1:def 4; hence h * e = h; e * h = 1_Gp2 * h9 by A2 .= h9 by GROUP_1:def 4; hence e * h = h; end; assume H99 = (1).Gp2; then the carrier of H99 = {1_Gp2} by GROUP_2:def 7; then the carrier of H = {1_G} by A8,GROUP_1:def 4; hence contradiction by A6,Def8; end; H99 <> (Omega).Gp2 by A2,A5,Lm5; hence contradiction by A7,Def12,Lm6; end; end; hence thesis; end; end; definition let O be set; let G be GroupWithOperators of O; let N be normal StableSubgroup of G; func Cosets N -> set means :Def14: for H being strict normal Subgroup of G st H = the multMagma of N holds it = Cosets H; existence proof reconsider H = the multMagma of N as strict normal Subgroup of G by Lm7; set x = Cosets H; take x; let H be strict normal Subgroup of G; assume H = the multMagma of N; hence thesis; end; uniqueness proof reconsider H = the multMagma of N as strict normal Subgroup of G by Lm7; let y1,y2 be set; assume for H being strict normal Subgroup of G st H = the multMagma of N holds y1 = Cosets H; then A1: y1 = Cosets H; assume for H being strict normal Subgroup of G st H = the multMagma of N holds y2 = Cosets H; hence thesis by A1; end; end; definition let O be set; let G be GroupWithOperators of O; let N be normal StableSubgroup of G; func CosOp N -> BinOp of Cosets N means :Def15: for H being strict normal Subgroup of G st H = the multMagma of N holds it = CosOp H; existence proof reconsider H = the multMagma of N as strict normal Subgroup of G by Lm7; Cosets N = Cosets H by Def14; then reconsider x = CosOp H as BinOp of Cosets N; take x; let H be strict normal Subgroup of G; assume H = the multMagma of N; hence thesis; end; uniqueness proof reconsider H = the multMagma of N as strict normal Subgroup of G by Lm7; let y1,y2 be BinOp of Cosets N; assume for H being strict normal Subgroup of G st H = the multMagma of N holds y1 = CosOp H; then A1: y1 = CosOp H; assume for H being strict normal Subgroup of G st H = the multMagma of N holds y2 = CosOp H; hence y1=y2 by A1; end; end; Lm9: for G being Group, N being normal Subgroup of G, A being Element of Cosets N, g being Element of G holds g in A iff A = g * N proof let G be Group; let N be normal Subgroup of G; let A be Element of Cosets N; let g be Element of G; hereby consider a be Element of G such that A1: A = a * N by GROUP_2:def 15; assume g in A; then consider h be Element of G such that A2: g = a*h and A3: h in N by A1,GROUP_2:103; g" * a = h"*a"*a by A2,GROUP_1:17 .= h"*(a"*a) by GROUP_1:def 3 .= h"*1_G by GROUP_1:def 5 .= h" by GROUP_1:def 4; then g" * a in N by A3,GROUP_2:51; hence A = g * N by A1,GROUP_2:114; end; g = g * 1_G & 1_G in N by GROUP_1:def 4,GROUP_2:46; hence thesis by GROUP_2:103; end; Lm10: for O being set, o being Element of O, G being GroupWithOperators of O, H being StableSubgroup of G, g being Element of G st g in H holds (G^o).g in H proof let O be set; let o be Element of O; let G be GroupWithOperators of O; let H be StableSubgroup of G; let g be Element of G; set f=G^o; assume g in H; then A1: g in the carrier of H by STRUCT_0:def 5; then f.g = (f|the carrier of H).g by FUNCT_1:49; then A2: f.g = (H^o).g by Def7; (H^o).g in the carrier of H by A1,FUNCT_2:5; hence thesis by A2,STRUCT_0:def 5; end; definition let O be set; let G be GroupWithOperators of O; let N be normal StableSubgroup of G; func CosAc N -> Action of O, Cosets N means :Def16: for o being Element of O holds it.o = {[A,B] where A,B is Element of Cosets N: ex g,h being Element of G st g in A & h in B & h = (G^o).g} if O is not empty otherwise it=[:{},{id Cosets N}:]; existence proof A1: now deffunc F(set) = {[A,B] where A,B is Element of Cosets N: for o being Element of O st $1=o holds ex g,h being Element of G st g in A & h in B & h = ( G^o).g}; reconsider H = the multMagma of N as strict normal Subgroup of G by Lm7; assume A2: O is not empty; A3: Cosets N = Cosets H by Def14; A4: now let x be set; set f=F(x); A5: now let y be set; assume y in f; then consider A,B be Element of Cosets N such that A6: y = [A,B] and for o being Element of O st x=o holds ex g,h being Element of G st g in A & h in B & h = (G^o).g; reconsider A,B as set; take A,B; thus y = [A,B] by A6; end; assume A7: x in O; now reconsider o=x as Element of O by A7; let y,y1,y2 be set; assume [y,y1] in f; then consider A1,B1 be Element of Cosets N such that A8: [y,y1] = [A1,B1] and A9: for o being Element of O st x=o holds ex g,h being Element of G st g in A1 & h in B1 & h = (G^o).g; assume [y,y2] in f; then consider A2,B2 be Element of Cosets N such that A10: [y,y2] = [A2,B2] and A11: for o being Element of O st x=o holds ex g,h being Element of G st g in A2 & h in B2 & h = (G^o).g; A12: y1=B1 by A8,XTUPLE_0:1; A13: y2=B2 by A10,XTUPLE_0:1; A14: y=A2 by A10,XTUPLE_0:1; set f=G^o; A15: y=A1 by A8,XTUPLE_0:1; consider g1,h1 be Element of G such that A16: g1 in A1 and A17: h1 in B1 and A18: h1 = (G^o).g1 by A9; consider g2,h2 be Element of G such that A19: g2 in A2 and A20: h2 in B2 and A21: h2 = (G^o).g2 by A11; reconsider A1,A2,B1,B2 as Element of Cosets H by Def14; A22: A2 = g2 * H by A19,Lm9; A1 = g1 * H by A16,Lm9; then g2" * g1 in H by A15,A14,A22,GROUP_2:114; then g2" * g1 in the carrier of H by STRUCT_0:def 5; then g2" * g1 in N by STRUCT_0:def 5; then f.(g2" * g1) in N by Lm10; then f.(g2") * f.g1 in N by GROUP_6:def 6; then h2" * h1 in N by A18,A21,GROUP_6:32; then h2" * h1 in the carrier of N by STRUCT_0:def 5; then A23: h2" * h1 in H by STRUCT_0:def 5; A24: B2 = h2 * H by A20,Lm9; B1 = h1 * H by A17,Lm9; hence y1 = y2 by A12,A13,A23,A24,GROUP_2:114; end; then reconsider f as Function by A5,FUNCT_1:def 1,RELAT_1:def 1; now let y1 be set; hereby reconsider o=x as Element of O by A7; assume A25: y1 in Cosets N; then reconsider A=y1 as Element of Cosets N; y1 in Cosets H by A25,Def14; then consider g be Element of G such that A26: y1 = g * H and y1 = H * g by GROUP_6:13; set h = (G^o).g; reconsider B = h * H as Element of Cosets N by A3,GROUP_2:def 15; reconsider y2=B as set; take y2; now let o be Element of O; assume A27: x=o; take g,h; thus g in A by A3,A26,Lm9; thus h in B by A3,Lm9; thus h = (G^o).g by A27; end; hence [y1,y2] in f; end; given y2 be set such that A28: [y1,y2] in f; consider A,B be Element of Cosets N such that A29: [y1,y2] = [A,B] and for o being Element of O st x=o holds ex g,h being Element of G st g in A & h in B & h = (G^o).g by A28; A in Cosets N by A3; hence y1 in Cosets N by A29,XTUPLE_0:1; end; then A30: dom f = Cosets N by XTUPLE_0:def 12; now let y2 be set; assume y2 in rng f; then consider y1 be set such that A31: [y1,y2] in f by XTUPLE_0:def 13; consider A,B be Element of Cosets N such that A32: [y1,y2] = [A,B] and for o being Element of O st x=o holds ex g,h being Element of G st g in A & h in B & h = (G^o).g by A31; B in Cosets N by A3; hence y2 in Cosets N by A32,XTUPLE_0:1; end; then rng f c= Cosets N by TARSKI:def 3; hence F(x) in Funcs(Cosets N,Cosets N) by A30,FUNCT_2:def 2; end; ex f being Function of O,Funcs(Cosets N,Cosets N) st for x being set st x in O holds f.x = F(x) from FUNCT_2:sch 2(A4); then consider IT be Function of O,Funcs(Cosets N,Cosets N) such that A33: for x being set st x in O holds IT.x = F(x); reconsider IT as Action of O, Cosets N; take IT; let o be Element of O; reconsider x=o as set; set X = {[A,B] where A,B is Element of Cosets N: for o being Element of O st x=o holds ex g,h being Element of G st g in A & h in B & h = (G^o).g}; set Y = {[A,B] where A,B is Element of Cosets N: ex g,h being Element of G st g in A & h in B & h = (G^o).g}; A34: now let y be set; hereby assume y in X; then consider A,B be Element of Cosets N such that A35: y = [A,B] and A36: for o being Element of O st x=o holds ex g,h being Element of G st g in A & h in B & h = (G^o).g; ex g,h being Element of G st g in A & h in B & h = (G^o).g by A36; hence y in Y by A35; end; assume y in Y; then consider A,B be Element of Cosets N such that A37: y = [A,B] and A38: ex g,h being Element of G st g in A & h in B & h = (G^o).g; for o being Element of O st x=o holds ex g,h being Element of G st g in A & h in B & h = (G^o).g by A38; hence y in X by A37; end; IT.o = X by A2,A33; hence IT.o = Y by A34,TARSKI:1; end; now assume O is empty; then reconsider IT=[:{},{id Cosets N}:] as Action of O, Cosets N by Lm2; take IT; thus IT=[:{},{id Cosets N}:]; end; hence thesis by A1; end; uniqueness proof now assume O is not empty; let IT1,IT2 be Action of O, Cosets N; assume A39: for o being Element of O holds IT1.o = {[A,B] where A,B is Element of Cosets N: ex g,h being Element of G st g in A & h in B & h = (G^o).g }; assume A40: for o being Element of O holds IT2.o = {[A,B] where A,B is Element of Cosets N: ex g,h being Element of G st g in A & h in B & h = (G^o).g }; A41: now let x be set; assume x in dom IT1; then reconsider o=x as Element of O; IT1.o = {[A,B] where A,B is Element of Cosets N: ex g,h being Element of G st g in A & h in B & h = (G^o).g} by A39; hence IT1.x=IT2.x by A40; end; dom IT1 = O & dom IT2 = O by FUNCT_2:def 1; hence IT1=IT2 by A41,FUNCT_1:2; end; hence thesis; end; correctness; end; definition let O be set; let G be GroupWithOperators of O; let N be normal StableSubgroup of G; func G./.N -> HGrWOpStr over O equals HGrWOpStr (# Cosets N, CosOp N, CosAc N #); correctness; end; registration let O be set; let G be GroupWithOperators of O; let N be normal StableSubgroup of G; cluster G./.N -> non empty; correctness proof reconsider H = the multMagma of N as strict normal Subgroup of G by Lm7; Cosets N = Cosets H by Def14; hence thesis; end; cluster G./.N -> distributive Group-like associative; correctness proof reconsider H = the multMagma of N as strict normal Subgroup of G by Lm7; set G9 = the multMagma of G./.N; A1: now set e9=1_(G./.H); reconsider e=e9 as Element of G./.N by Def14; take e; let h be Element of G./.N; reconsider h9=h as Element of G./.H by Def14; set g=h9"; set g9=g; h*e = h9*e9 by Def15 .= h9 by GROUP_1:def 4; hence h * e = h; e*h = e9*h9 by Def15 .= h9 by GROUP_1:def 4; hence e * h = h; reconsider g as Element of G./.N by Def14; take g; h*g = h9*g9 by Def15 .= 1_(G./.H) by GROUP_1:def 5; hence h * g = e; g*h = g9*h9 by Def15 .= 1_(G./.H) by GROUP_1:def 5; hence g * h = e; end; A2: now let G9 be Group; let a be Action of O, the carrier of G9; assume A3: a = the action of G./.N; assume A4: the multMagma of G9 = the multMagma of G./.N; now let o be Element of O; assume A5: o in O; then A6: a.o = {[A,B] where A,B is Element of Cosets N: ex g,h being Element of G st g in A & h in B & h = (G^o).g} by A3,Def16; a.o in Funcs(Cosets N, Cosets N) by A3,A5,FUNCT_2:5; then consider f be Function such that A7: a.o = f and A8: dom f = Cosets N and A9: rng f c= Cosets N by FUNCT_2:def 2; reconsider f as Function of the carrier of G9,the carrier of G9 by A4 ,A8,A9,FUNCT_2:2; now let A1,A2 be Element of G9; set A3=A1*A2; set B1=f.A1,B2=f.A2,B3=f.A3; [A1,B1] in f by A4,A8,FUNCT_1:1; then consider A19,B19 be Element of Cosets N such that A10: [A1,B1] = [A19,B19] and A11: ex g1,h1 being Element of G st g1 in A19 & h1 in B19 & h1 = (G^o). g1 by A6,A7; [A2,B2] in f by A4,A8,FUNCT_1:1; then consider A29,B29 be Element of Cosets N such that A12: [A2,B2] = [A29,B29] and A13: ex g2,h2 being Element of G st g2 in A29 & h2 in B29 & h2 = (G^o). g2 by A6,A7; [A3,B3] in f by A4,A8,FUNCT_1:1; then consider A39,B39 be Element of Cosets N such that A14: [A3,B3] = [A39,B39] and A15: ex g3,h3 being Element of G st g3 in A39 & h3 in B39 & h3 = (G^o). g3 by A6,A7; consider g3,h3 be Element of G such that A16: g3 in A39 and A17: h3 in B39 and A18: h3 = (G^o).g3 by A15; consider g2,h2 be Element of G such that A19: g2 in A29 and A20: h2 in B29 and A21: h2 = (G^o).g2 by A13; consider g1,h1 be Element of G such that A22: g1 in A19 and A23: h1 in B19 and A24: h1 = (G^o).g1 by A11; A25: @((nat_hom H).g1)=(nat_hom H).g1 & @((nat_hom H).g2)=(nat_hom H ).g2; A26: (nat_hom H).g1=g1*H & (nat_hom H).g2=g2*H by GROUP_6:def 8; reconsider A19,A29,A39,B19,B29,B39 as Element of Cosets H by Def14; A27: A29 = g2 * H by A19,Lm9; A28: A39 = g3 * H by A16,Lm9; A29: B29 = h2 * H by A20,Lm9; reconsider A19,A29,B19,B29 as Element of G./.H; A2=g2 * H by A12,A27,XTUPLE_0:1; then A1*A2 = (the multF of G9).(A19,A29) by A10,A27,XTUPLE_0:1 .= @(A19*A29) by A4,Def15 .= @A19 * @A29 by GROUP_6:20; then A1*A2 = (g1 * H)*(g2 * H) by A22,A27,Lm9 .= ((nat_hom H).g1)*((nat_hom H).g2) by A25,A26,GROUP_6:19 .= (nat_hom H).(g1*g2) by GROUP_6:def 6 .= (g1*g2) * H by GROUP_6:def 8; then g3 * H = (g1*g2) * H by A14,A28,XTUPLE_0:1; then g3" * (g1*g2) in H by GROUP_2:114; then g3" * (g1*g2) in the carrier of H by STRUCT_0:def 5; then g3" * (g1*g2) in N by STRUCT_0:def 5; then (G^o).(g3" * (g1*g2)) in N by Lm10; then (G^o).(g3") * ((G^o).(g1*g2)) in N by GROUP_6:def 6; then (G^o).(g3") * (((G^o).g1)*(G^o).g2) in N by GROUP_6:def 6; then h3" * (h1*h2) in N by A24,A21,A18,GROUP_6:32; then h3" * (h1*h2) in the carrier of N by STRUCT_0:def 5; then A30: h3" * (h1*h2) in H by STRUCT_0:def 5; A31: (nat_hom H).h1=h1*H & (nat_hom H).h2=h2*H by GROUP_6:def 8; B39 = h3 * H by A17,Lm9; then A32: B3=h3 * H by A14,XTUPLE_0:1; A33: @((nat_hom H).h1)=(nat_hom H).h1 & @((nat_hom H).h2)=(nat_hom H ).h2; B2=h2 * H by A12,A29,XTUPLE_0:1; then B1*B2 = (the multF of G9).(B19,B29) by A10,A29,XTUPLE_0:1 .= @(B19*B29) by A4,Def15 .= @B19 * @B29 by GROUP_6:20; then B1*B2 = (h1 * H)*(h2 * H) by A23,A29,Lm9 .= ((nat_hom H).h1)*((nat_hom H).h2) by A33,A31,GROUP_6:19 .= (nat_hom H).(h1*h2) by GROUP_6:def 6 .= (h1*h2) * H by GROUP_6:def 8; hence f.A3 = f.A1 * f.A2 by A32,A30,GROUP_2:114; end; hence a.o is Homomorphism of G9,G9 by A7,GROUP_6:def 6; end; hence a is distributive by Def4; end; the carrier of G./.N = the carrier of G./.H by Def14; then A34: G9 is Group-like associative by Def15; now let x,y,z be Element of G./.N; reconsider x9=x,y9=y,z9=z as Element of G9; (x9*y9)*z9 = (x*y)*z & x9*(y9*z9) = x*(y*z); hence (x*y)*z = x*(y*z) by A34,GROUP_1:def 3; end; hence thesis by A1,A2,Def5,GROUP_1:def 2,def 3; end; end; :: ALG I.4.2 Definition 3 definition let O be set; let G,H be GroupWithOperators of O; let f be Function of G, H; attr f is homomorphic means :Def18: for o being Element of O, g being Element of G holds f.((G^o).g) = (H^o).(f.g); end; registration let O be set; let G,H be GroupWithOperators of O; cluster multiplicative homomorphic for Function of G, H; existence proof set f = 1:(G,H); reconsider f as Function of G, H; take f; thus f is multiplicative; let o be Element of O; let g be Element of G; (H^o).(f.g) = (H^o).(1_H) by GROUP_6:def 7 .= 1_H by GROUP_6:31; hence thesis by GROUP_6:def 7; end; end; definition let O be set; let G,H be GroupWithOperators of O; mode Homomorphism of G, H is multiplicative homomorphic Function of G, H; end; :: like GROUP_6:48 Lm11: for O being set, G,H,I being GroupWithOperators of O, h being Homomorphism of G, H for h1 being Homomorphism of H, I holds h1 * h is Homomorphism of G,I proof let O be set; let G,H,I be GroupWithOperators of O; let h be Homomorphism of G, H; let h1 be Homomorphism of H, I; reconsider f = h1 * h as Function of G, I; now let o be Element of O; let g be Element of G; thus f.((G^o).g) = h1.(h.((G^o).g)) by FUNCT_2:15 .= h1.((H^o).(h.g)) by Def18 .= (I^o).(h1.(h.g)) by Def18 .= (I^o).(f.g) by FUNCT_2:15; end; hence thesis by Def18; end; definition let O be set; let G,H,I be GroupWithOperators of O; let h be Homomorphism of G, H; let h1 be Homomorphism of H, I; redefine func h1 * h -> Homomorphism of G,I; correctness by Lm11; end; definition let O be set; let G,H be GroupWithOperators of O; pred G,H are_isomorphic means :Def19: ex h being Homomorphism of G,H st h is bijective; reflexivity proof let G be GroupWithOperators of O; reconsider G9=G as Group; set h = id the carrier of G9; now let o be Element of O; let g be Element of G; h.((G^o).g) = (G^o).g by FUNCT_1:17; hence h.((G^o).g) = (G^o).(h.g) by FUNCT_1:17; end; then reconsider h as Homomorphism of G, G by Def18,GROUP_6:38; take h; rng h = the carrier of G by RELAT_1:45; then h is onto by FUNCT_2:def 3; hence thesis; end; end; :: like GROUP_6:77 Lm12: for O being set, G,H being GroupWithOperators of O holds G,H are_isomorphic implies H,G are_isomorphic proof let O be set; let G,H be GroupWithOperators of O; assume G,H are_isomorphic; then consider f be Homomorphism of G,H such that A1: f is bijective by Def19; set f9 = f"; A2: rng f = the carrier of H by A1,FUNCT_2:def 3; then A3: dom f9 = the carrier of H by A1,FUNCT_1:33; A4: dom f = the carrier of G by FUNCT_2:def 1; then A5: rng f9 = the carrier of G by A1,FUNCT_1:33; then reconsider f9 as Function of H,G by A3,FUNCT_2:1; A6: now let o be Element of O; let h be Element of H; set g=f9.h; thus f9.((H^o).h) = f9.((H^o).(f.g)) by A1,A2,FUNCT_1:35 .= f9.(f.((G^o).g)) by Def18 .= (G^o).(f9.h) by A1,A4,FUNCT_1:34; end; now let h1,h2 be Element of H; set g1=f9.h1; set g2=f9.h2; f.g1=h1 & f.g2=h2 by A1,A2,FUNCT_1:35; hence f9.(h1*h2) = f9.(f.(g1*g2)) by GROUP_6:def 6 .= f9.h1 * f9.h2 by A1,A4,FUNCT_1:34; end; then reconsider f9 as Homomorphism of H,G by A6,Def18,GROUP_6:def 6; take f9; f9 is onto by A5,FUNCT_2:def 3; hence thesis by A1; end; definition let O be set, G,H be GroupWithOperators of O; redefine pred G,H are_isomorphic; symmetry by Lm12; end; definition let O be set; let G be GroupWithOperators of O; let N be normal StableSubgroup of G; func nat_hom N -> Homomorphism of G, G./.N means :Def20: for H being strict normal Subgroup of G st H = the multMagma of N holds it = nat_hom H; existence proof set H = the multMagma of N; reconsider H as strict normal Subgroup of G by Lm7; set IT = nat_hom H; reconsider K = G./.N as GroupWithOperators of O; reconsider IT9 = IT as Function of G, K by Def14; A1: now let a, b be Element of G; IT9.(a * b) = IT.a * IT.b by GROUP_6:def 6 .= IT9.a * IT9.b by Def15; hence IT9.(a * b) = IT9.a * IT9.b; end; now let o be Element of O; let g be Element of G; per cases; suppose A2: O<>{}; then (the action of K).o in Funcs(the carrier of K, the carrier of K) by FUNCT_2:5; then consider f be Function such that A3: f=(the action of K).o and A4: dom f = the carrier of K and rng f c= the carrier of K by FUNCT_2:def 2; A5: f = {[A,B] where A,B is Element of Cosets N: ex g,h being Element of G st g in A & h in B & h = (G^o).g} by A2,A3,Def16; [IT9.g, f.(IT9.g)] in f by A4,FUNCT_1:def 2; then consider A,B be Element of Cosets N such that A6: [IT9.g, f.(IT9.g)] = [A,B] and A7: ex g,h being Element of G st g in A & h in B & h = (G^o).g by A5; A8: IT9.g = A by A6,XTUPLE_0:1; consider g9,h9 be Element of G such that A9: g9 in A and A10: h9 in B & h9 = (G^o).g9 by A7; A11: (G^o).(g9" * g) = (G^o).(g9") * (G^o).g by GROUP_6:def 6 .= ((G^o).g9)" * (G^o).g by GROUP_6:32; reconsider A,B as Element of Cosets H by Def14; A = g9 * H by A9,Lm9; then g * H = g9 * H by A8,GROUP_6:def 8; then g9" * g in H by GROUP_2:114; then g9" * g in the carrier of N by STRUCT_0:def 5; then g9" * g in N by STRUCT_0:def 5; then (G^o).(g9" * g) in N by Lm10; then (G^o).(g9" * g) in the carrier of N by STRUCT_0:def 5; then A12: (G^o).(g9" * g) in H by STRUCT_0:def 5; A13: (K^o).(IT9.g) = f.(IT9.g) by A2,A3,Def6; IT9.((G^o).g) = ((G^o).g) * H by GROUP_6:def 8 .= ((G^o).g9) * H by A12,A11,GROUP_2:114 .= B by A10,Lm9; hence IT9.((G^o).g) = (K^o).(IT9.g) by A13,A6,XTUPLE_0:1; end; suppose A14: O={}; then G^o = id the carrier of G by Def6; then A15: (G^o).g = g by FUNCT_1:18; K^o = id the carrier of K by A14,Def6; hence IT9.((G^o).g) = (K^o).(IT9.g) by A15,FUNCT_1:18; end; end; then reconsider IT9 as Homomorphism of G, K by A1,Def18,GROUP_6:def 6; reconsider IT9 as Homomorphism of G, G./.N; take IT9; let H be strict normal Subgroup of G; assume H = the multMagma of N; hence thesis; end; uniqueness proof reconsider H = the multMagma of N as strict normal Subgroup of G by Lm7; let IT1,IT2 be Homomorphism of G, G./.N; assume for H being strict normal Subgroup of G st H = the multMagma of N holds IT1 = nat_hom H; then A16: IT1 = nat_hom H; assume for H being strict normal Subgroup of G st H = the multMagma of N holds IT2 = nat_hom H; hence thesis by A16; end; end; :: like GROUP_6:40 Lm13: for O being set, G,H being GroupWithOperators of O, g being Homomorphism of G,H holds g.(1_G)=1_H proof let O be set; let G,H be GroupWithOperators of O; let g be Homomorphism of G,H; g.(1_G) = g.(1_G * 1_G) by GROUP_1:def 4 .= g.(1_G) * g.(1_G) by GROUP_6:def 6; hence thesis by GROUP_1:7; end; :: like GROUP_6:41 Lm14: for O being set, G,H being GroupWithOperators of O, a being Element of G , g being Homomorphism of G,H holds g.(a")=(g.a)" proof let O be set; let G,H be GroupWithOperators of O; let a be Element of G; let g be Homomorphism of G,H; g.(a") * g.a = g.(a" * a) by GROUP_6:def 6 .= g.(1_G) by GROUP_1:def 5 .= 1_H by Lm13; hence thesis by GROUP_1:12; end; :: like GROUP_2:61 Lm15: for O being set, G being GroupWithOperators of O, A being Subset of G st A <> {} & (for g1,g2 being Element of G st g1 in A & g2 in A holds g1 * g2 in A ) & (for g being Element of G st g in A holds g" in A) & (for o being Element of O, g being Element of G st g in A holds (G^o).g in A) holds ex H being strict StableSubgroup of G st the carrier of H = A proof let O be set; let G be GroupWithOperators of O; let A be Subset of G; assume A1: A <> {}; assume A2: for g1,g2 being Element of G st g1 in A & g2 in A holds g1 * g2 in A; assume for g being Element of G st g in A holds g" in A; then consider H9 be strict Subgroup of G such that A3: the carrier of H9 = A by A1,A2,GROUP_2:52; set m9 = the multF of H9; set A9 = the carrier of H9; assume A4: for o being Element of O, g being Element of G st g in A holds (G^o) .g in A; A5: now let H be non empty HGrWOpStr over O; let a9 be Action of O,A9; assume A6: H = HGrWOpStr (#A9,m9,a9#); now let x,y,z be Element of H; reconsider x9=x,y9=y,z9=z as Element of H9 by A6; (x*y)*z = (x9*y9)*z9 by A6 .= x9*(y9*z9) by GROUP_1:def 3; hence (x*y)*z = x*(y*z) by A6; end; hence H is associative by GROUP_1:def 3; now set e9=1_H9; reconsider e=e9 as Element of H by A6; take e; let h be Element of H; reconsider h9=h as Element of H9 by A6; set g9=h9"; h*e = h9*e9 by A6 .= h9 by GROUP_1:def 4; hence h * e = h; e*h = e9*h9 by A6 .= h9 by GROUP_1:def 4; hence e * h = h; reconsider g=g9 as Element of H by A6; take g; h*g = h9*g9 by A6 .= 1_H9 by GROUP_1:def 5; hence h * g = e; g*h = g9*h9 by A6 .= 1_H9 by GROUP_1:def 5; hence g * h = e; end; hence H is Group-like by GROUP_1:def 2; end; per cases; suppose A7: O is empty; set a9 = [:{},{id A9}:]; reconsider a9 as Action of O,A9 by A7,Lm2; set H = HGrWOpStr (#A9,m9,a9#); reconsider H as non empty HGrWOpStr over O; now let G9 be Group; let a be Action of O, the carrier of G9; assume a = the action of H; assume the multMagma of G9 = the multMagma of H; for o being Element of O st o in O holds a.o is Homomorphism of G9, G9 by A7; hence a is distributive by Def4; end; then reconsider H as GroupWithOperators of O by A5,Def5; A8: the carrier of H c= the carrier of G by GROUP_2:def 5; A9: now let o be Element of O; A10: now let x,y be set; assume A11: [x,y] in (id the carrier of G)|the carrier of H; then [x,y] in id the carrier of G by RELAT_1:def 11; then A12: x=y by RELAT_1:def 10; x in the carrier of H by A11,RELAT_1:def 11; hence [x,y] in id the carrier of H by A12,RELAT_1:def 10; end; A13: now let x,y be set; assume A14: [x,y] in id the carrier of H; then A15: x in the carrier of H by RELAT_1:def 10; x=y by A14,RELAT_1:def 10; then [x,y] in id the carrier of G by A8,A15,RELAT_1:def 10; hence [x,y] in (id the carrier of G)|the carrier of H by A15, RELAT_1:def 11; end; H^o = id the carrier of H by A7,Def6 .= (id the carrier of G)|the carrier of H by A13,A10,RELAT_1:def 2; hence H^o = (G^o)|the carrier of H by A7,Def6; end; the multF of H = (the multF of G)||the carrier of H by GROUP_2:def 5; then H is Subgroup of G by A8,GROUP_2:def 5; then reconsider H as strict StableSubgroup of G by A9,Def7; take H; thus thesis by A3; end; suppose A16: O is not empty; set a9 = {[o,(G^o)|A9] where o is Element of O : not contradiction}; now let x be set; assume x in a9; then ex o be Element of O st x=[o,(G^o)|A9]; hence ex y1,y2 being set st x = [y1,y2]; end; then reconsider a9 as Relation by RELAT_1:def 1; A17: now let x be set; assume x in O; then reconsider o=x as Element of O; reconsider y = (G^o)|A9 as set; take y; thus [x,y] in a9; end; now let x be set; given y be set such that A18: [x,y] in a9; consider o be Element of O such that A19: [x,y] = [o,(G^o)|A9] by A18; o in O by A16; hence x in O by A19,XTUPLE_0:1; end; then A20: dom a9 = O by A17,XTUPLE_0:def 12; now let x,y1,y2 be set; assume [x,y1] in a9; then consider o1 be Element of O such that A21: [x,y1]=[o1,(G^o1)|A9]; A22: x=o1 by A21,XTUPLE_0:1; assume [x,y2] in a9; then consider o2 be Element of O such that A23: [x,y2]=[o2,(G^o2)|A9]; x=o2 by A23,XTUPLE_0:1; hence y1 = y2 by A21,A23,A22,XTUPLE_0:1; end; then reconsider a9 as Function by FUNCT_1:def 1; now let y be set; assume y in rng a9; then consider x be set such that A24: x in dom a9 & y = a9.x by FUNCT_1:def 3; [x,y] in a9 by A24,FUNCT_1:1; then consider o be Element of O such that A25: [x,y]=[o,(G^o)|A9]; now reconsider f = (G^o)|A9 as Function; take f; A26: dom((G^o)|A9) = dom((G^o)*(id A9)) by RELAT_1:65 .= dom(G^o) /\ A9 by FUNCT_1:19 .= (the carrier of G) /\ A9 by FUNCT_2:def 1; thus y = f by A25,XTUPLE_0:1; A9 c= the carrier of G by GROUP_2:def 5; hence A27: dom f = A9 by A26,XBOOLE_1:28; now let y be set; A28: dom f = dom((G^o)*(id A9)) by RELAT_1:65; assume y in rng f; then consider x be set such that A29: x in dom f and A30: y = f.x by FUNCT_1:def 3; y = ((G^o)*(id A9)).x by A30,RELAT_1:65 .= (G^o).((id A9).x) by A28,A29,FUNCT_1:12 .= (G^o).x by A27,A29,FUNCT_1:18; hence y in A9 by A4,A3,A27,A29; end; hence rng f c= A9 by TARSKI:def 3; end; hence y in Funcs(A9,A9) by FUNCT_2:def 2; end; then rng a9 c= Funcs(A9,A9) by TARSKI:def 3; then reconsider a9 as Action of O,A9 by A20,FUNCT_2:2; reconsider H = HGrWOpStr (#A9,m9,a9#) as non empty HGrWOpStr over O; A31: the multF of H = (the multF of G)||the carrier of H by GROUP_2:def 5; H is Group-like & the carrier of H c= the carrier of G by A5,GROUP_2:def 5; then A32: H is Subgroup of G by A31,GROUP_2:def 5; now let G9 be Group; let a be Action of O, the carrier of G9; assume A33: a = the action of H; assume A34: the multMagma of G9 = the multMagma of H; now let o be Element of O; assume o in O; then A35: o in dom a by FUNCT_2:def 1; then a.o in rng a by FUNCT_1:3; then consider f be Function such that A36: a.o = f and A37: dom f = the carrier of G9 & rng f c= the carrier of G9 by FUNCT_2:def 2 ; reconsider f as Function of G9,G9 by A37,FUNCT_2:2; [o,a.o] in a9 by A33,A35,FUNCT_1:1; then consider o9 be Element of O such that A38: [o,a.o] = [o9,(G^o9)|A9]; A39: o=o9 & a.o=(G^o9)|A9 by A38,XTUPLE_0:1; now let a9, b9 be Element of G9; b9 in the carrier of H9 by A34; then A40: b9 in dom id A9; reconsider a=a9,b=b9 as Element of H by A34; reconsider g1=a,g2=b as Element of G by GROUP_2:42; a9 in the carrier of H9 by A34; then A41: a9 in dom id A9; reconsider h1=(G^o).g1,h2=(G^o).g2 as Element of H by A4,A3; a9*b9 in the carrier of H9 by A34; then A42: a9*b9 in dom id A9; A43: f.b9 = ((G^o)*(id A9)).b9 by A36,A39,RELAT_1:65 .= (G^o).((id A9).b9) by A40,FUNCT_1:13 .= h2 by FUNCT_1:18; A44: f.a9 = ((G^o)*(id A9)).a9 by A36,A39,RELAT_1:65 .= (G^o).((id A9).a9) by A41,FUNCT_1:13 .= h1 by FUNCT_1:18; thus f.(a9 * b9) = ((G^o)*(id A9)).(a9*b9) by A36,A39,RELAT_1:65 .= (G^o).((id A9).(a9*b9)) by A42,FUNCT_1:13 .= (G^o).(a*b) by A34,FUNCT_1:18 .= (G^o).(g1*g2) by A32,GROUP_2:43 .= ((G^o).g1) * ((G^o).g2) by GROUP_6:def 6 .= h1 * h2 by A32,GROUP_2:43 .= f.a9 * f.b9 by A34,A44,A43; end; hence a.o is Homomorphism of G9,G9 by A36,GROUP_6:def 6; end; hence a is distributive by Def4; end; then reconsider H as GroupWithOperators of O by A5,Def5; now let o be Element of O; o in O by A16; then o in dom a9 by FUNCT_2:def 1; then [o,a9.o] in a9 by FUNCT_1:1; then consider o9 be Element of O such that A45: [o,a9.o] = [o9,(G^o9)|A9]; o=o9 & a9.o=(G^o9)|A9 by A45,XTUPLE_0:1; hence H^o = (G^o)|the carrier of H by A16,Def6; end; then reconsider H as strict StableSubgroup of G by A32,Def7; take H; thus thesis by A3; end; end; definition let O be set; let G,H be GroupWithOperators of O; let g be Homomorphism of G, H; func Ker g -> strict StableSubgroup of G means :Def21: the carrier of it = { a where a is Element of G: g.a = 1_H}; existence proof defpred P[set] means g.$1 = 1_H; reconsider A = {a where a is Element of G: P[a]} as Subset of G from DOMAIN_1:sch 7; A1: now let a,b be Element of G; assume a in A & b in A; then A2: ( ex a1 being Element of G st a1 = a & g.a1 = 1_H)& ex b1 being Element of G st b1 = b & g.b1 = 1_H; g.(a * b) = g.a * g.b by GROUP_6:def 6 .= 1_H by A2,GROUP_1:def 4; hence a * b in A; end; A3: now let a be Element of G; assume a in A; then ex a1 being Element of G st a1 = a & g.a1 = 1_H; then g.(a") = (1_H)" by Lm14 .= 1_H by GROUP_1:8; hence a" in A; end; A4: now let o be Element of O; let a be Element of G; assume a in A; then ex a1 being Element of G st a1 = a & g.a1 = 1_H; then g.((G^o).a) = (H^o).(1_H) by Def18 .= 1_H by GROUP_6:31; hence (G^o).a in A; end; g.(1_G) = 1_H by Lm13; then 1_G in A; then consider B be strict StableSubgroup of G such that A5: the carrier of B = A by A1,A3,A4,Lm15; take B; thus thesis by A5; end; uniqueness by Lm5; end; registration let O be set; let G,H be GroupWithOperators of O; let g be Homomorphism of G, H; cluster Ker g -> normal; correctness proof now reconsider G9=G,H9=H as Group; let N be strict Subgroup of G; reconsider g9=g as Homomorphism of G9, H9; A1: the carrier of Ker g9 = {a where a is Element of G: g.a = 1_H} by GROUP_6:def 9; assume N = the multMagma of Ker g; then the carrier of Ker g9 = the carrier of N by A1,Def21; hence N is normal by GROUP_2:59; end; hence thesis by Def10; end; end; Lm16: for O being set, G being GroupWithOperators of O, H being StableSubgroup of G holds the multMagma of H is strict Subgroup of G proof let O be set; let G be GroupWithOperators of O; let H be StableSubgroup of G; reconsider H9=the multMagma of H as non empty multMagma; now set e=1_H; reconsider e9=e as Element of H9; take e9; let h9 be Element of H9; reconsider h=h9 as Element of H; set g=h"; reconsider g9=g as Element of H9; h9*e9 = h*e .= h by GROUP_1:def 4; hence h9 * e9 = h9; e9*h9 = e*h .= h by GROUP_1:def 4; hence e9 * h9 = h9; take g9; h9*g9 = h*g .= 1_H by GROUP_1:def 5; hence h9 * g9 = e9; g9*h9 = g*h .= 1_H by GROUP_1:def 5; hence g9 * h9 = e9; end; then reconsider H9 as Group-like non empty multMagma by GROUP_1:def 2; H is Subgroup of G by Def7; then the carrier of H9 c= the carrier of G & the multF of H9 = (the multF of G)|| the carrier of H9 by GROUP_2:def 5; hence thesis by GROUP_2:def 5; end; Lm17: for O being set, G,H being GroupWithOperators of O, G9 being strict StableSubgroup of G, f being Homomorphism of G,H holds ex H9 being strict StableSubgroup of H st the carrier of H9 = f.:(the carrier of G9) proof let O be set; let G,H be GroupWithOperators of O; let G9 be strict StableSubgroup of G; reconsider G99 = the multMagma of G9 as strict Subgroup of G by Lm16; let f be Homomorphism of G,H; set A = {f.g where g is Element of G:g in G99}; 1_G in G99 by GROUP_2:46; then f.(1_G) in A; then reconsider A as non empty set; now let x be set; assume x in A; then ex g be Element of G st x=f.g & g in G99; hence x in the carrier of H; end; then reconsider A as Subset of H by TARSKI:def 3; A1: now let h1,h2 be Element of H; assume that A2: h1 in A and A3: h2 in A; consider a be Element of G such that A4: h1=f.a & a in G99 by A2; consider b be Element of G such that A5: h2=f.b & b in G99 by A3; f.(a*b) = h1*h2 & a*b in G99 by A4,A5,GROUP_2:50,GROUP_6:def 6; hence h1*h2 in A; end; A6: now let o be Element of O; let h be Element of H; assume h in A; then consider g be Element of G such that A7: h=f.g and A8: g in G99; g in the carrier of G99 by A8,STRUCT_0:def 5; then g in G9 by STRUCT_0:def 5; then (G^o).g in G9 by Lm10; then (G^o).g in the carrier of G9 by STRUCT_0:def 5; then A9: (G^o).g in G99 by STRUCT_0:def 5; (H^o).h = f.((G^o).g) by A7,Def18; hence (H^o).h in A by A9; end; now let h be Element of H; assume h in A; then consider g be Element of G such that A10: h=f.g & g in G99; g" in G99 & h" = f.(g") by A10,Lm14,GROUP_2:51; hence h" in A; end; then consider H99 be strict StableSubgroup of H such that A11: the carrier of H99 = A by A1,A6,Lm15; take H99; now set R = f; let h be Element of H; reconsider R as Relation of the carrier of G, the carrier of H; hereby assume h in A; then consider g be Element of G such that A12: h=f.g and A13: g in G99; A14: g in the carrier of G9 by A13,STRUCT_0:def 5; dom f = the carrier of G by FUNCT_2:def 1; then [g,h] in f by A12,FUNCT_1:1; hence h in f.:(the carrier of G9) by A14,RELSET_1:29; end; assume h in f.:(the carrier of G9); then consider g be Element of G such that A15: [g,h] in R & g in the carrier of G9 by RELSET_1:29; f.g=h & g in G99 by A15,FUNCT_1:1,STRUCT_0:def 5; hence h in A; end; hence thesis by A11,SUBSET_1:3; end; definition let O be set; let G,H be GroupWithOperators of O; let g be Homomorphism of G, H; func Image g -> strict StableSubgroup of H means :Def22: the carrier of it = g.:(the carrier of G); existence proof reconsider G9 = the HGrWOpStr of G as strict StableSubgroup of G by Lm4; consider H9 be strict StableSubgroup of H such that A1: the carrier of H9 = g.:(the carrier of G9) by Lm17; take H9; thus thesis by A1; end; uniqueness by Lm5; end; definition let O be set; let G be GroupWithOperators of O; let H be StableSubgroup of G; func carr(H) -> Subset of G equals the carrier of H; coherence proof reconsider H9=H as Subgroup of G by Def7; carr(H9) is Subset of G; hence thesis; end; end; definition let O be set; let G be GroupWithOperators of O; let H1,H2 be StableSubgroup of G; func H1 * H2 -> Subset of G equals carr H1 * carr H2; coherence; end; :: like GROUP_2:55 Lm18: for O being set, G being GroupWithOperators of O, H being StableSubgroup of G holds 1_G in H proof let O be set; let G be GroupWithOperators of O; let H be StableSubgroup of G; H is Subgroup of G by Def7; hence thesis by GROUP_2:46; end; :: like GROUP_2:59 Lm19: for O being set, G being GroupWithOperators of O, H being StableSubgroup of G, g1,g2 being Element of G holds g1 in H & g2 in H implies g1 * g2 in H proof let O be set; let G be GroupWithOperators of O; let H be StableSubgroup of G; let g1,g2 be Element of G; assume A1: g1 in H & g2 in H; H is Subgroup of G by Def7; hence thesis by A1,GROUP_2:50; end; :: like GROUP_2:60 Lm20: for O being set, G being GroupWithOperators of O, H being StableSubgroup of G, g being Element of G holds g in H implies g" in H proof let O be set; let G be GroupWithOperators of O; let H be StableSubgroup of G; let g be Element of G; assume A1: g in H; H is Subgroup of G by Def7; hence thesis by A1,GROUP_2:51; end; definition let O be set; let G be GroupWithOperators of O; let H1,H2 be StableSubgroup of G; func H1 /\ H2 -> strict StableSubgroup of G means :Def25: the carrier of it = carr(H1) /\ carr(H2); existence proof set A = carr(H1) /\ carr(H2); 1_G in H2 by Lm18; then A1: 1_G in the carrier of H2 by STRUCT_0:def 5; A2: now let g1,g2 be Element of G; assume that A3: g1 in A and A4: g2 in A; g2 in carr(H2) by A4,XBOOLE_0:def 4; then A5: g2 in H2 by STRUCT_0:def 5; g1 in carr(H2) by A3,XBOOLE_0:def 4; then g1 in H2 by STRUCT_0:def 5; then g1 * g2 in H2 by A5,Lm19; then A6: g1 * g2 in carr(H2) by STRUCT_0:def 5; g2 in carr(H1) by A4,XBOOLE_0:def 4; then A7: g2 in H1 by STRUCT_0:def 5; g1 in carr(H1) by A3,XBOOLE_0:def 4; then g1 in H1 by STRUCT_0:def 5; then g1 * g2 in H1 by A7,Lm19; then g1 * g2 in carr(H1) by STRUCT_0:def 5; hence g1 * g2 in A by A6,XBOOLE_0:def 4; end; A8: now let o be Element of O; let a be Element of G; assume A9: a in A; then a in carr(H2) by XBOOLE_0:def 4; then a in H2 by STRUCT_0:def 5; then (G^o).a in H2 by Lm10; then A10: (G^o).a in carr(H2) by STRUCT_0:def 5; a in carr(H1) by A9,XBOOLE_0:def 4; then a in H1 by STRUCT_0:def 5; then (G^o).a in H1 by Lm10; then (G^o).a in carr(H1) by STRUCT_0:def 5; hence (G^o).a in A by A10,XBOOLE_0:def 4; end; A11: now let g be Element of G; assume A12: g in A; then g in carr(H2) by XBOOLE_0:def 4; then g in H2 by STRUCT_0:def 5; then g" in H2 by Lm20; then A13: g" in carr(H2) by STRUCT_0:def 5; g in carr(H1) by A12,XBOOLE_0:def 4; then g in H1 by STRUCT_0:def 5; then g" in H1 by Lm20; then g" in carr(H1) by STRUCT_0:def 5; hence g" in A by A13,XBOOLE_0:def 4; end; 1_G in H1 by Lm18; then 1_G in the carrier of H1 by STRUCT_0:def 5; then A <> {} by A1,XBOOLE_0:def 4; hence thesis by A2,A11,A8,Lm15; end; uniqueness by Lm5; commutativity; end; :: like GROUP_2:66 Lm21: for O being set, G being GroupWithOperators of O, H1,H2 being StableSubgroup of G holds the carrier of H1 c= the carrier of H2 implies H1 is StableSubgroup of H2 proof let O be set; let G be GroupWithOperators of O; let H1,H2 be StableSubgroup of G; reconsider H19=H1,H29=H2 as Subgroup of G by Def7; assume A1: the carrier of H1 c= the carrier of H2; A2: now let o be Element of O; thus H1^o = (G^o)|the carrier of H1 by Def7 .= ((G^o)|the carrier of H2)|the carrier of H1 by A1,RELAT_1:74 .= (H2^o)|the carrier of H1 by Def7; end; H19 is Subgroup of H29 by A1,GROUP_2:57; hence thesis by A2,Def7; end; :: like GROUP_4:def 5 definition let O be set; let G be GroupWithOperators of O; let A be Subset of G; func the_stable_subgroup_of A -> strict StableSubgroup of G means :Def26: A c= the carrier of it & for H being strict StableSubgroup of G st A c= the carrier of H holds it is StableSubgroup of H; existence proof defpred P[set] means ex H being strict StableSubgroup of G st $1 = carr H & A c= $1; consider X be set such that A1: for Y being set holds Y in X iff Y in bool the carrier of G & P[Y] from XBOOLE_0:sch 1; set M = meet X; A2: carr (Omega).G = the carrier of (Omega).G; then A3: X <> {} by A1; A4: the carrier of G in X by A1,A2; A5: M c= the carrier of G proof let y be set; consider x be set such that A6: x in X by A4; consider H be strict StableSubgroup of G such that A7: x = carr H and A c= x by A1,A6; assume y in M; then y in carr H by A6,A7,SETFAM_1:def 1; hence thesis; end; now let Y be set; assume Y in X; then consider H be strict StableSubgroup of G such that A8: Y = carr H and A c= Y by A1; 1_G in H by Lm18; hence 1_G in Y by A8,STRUCT_0:def 5; end; then A9: M <> {} by A3,SETFAM_1:def 1; reconsider M as Subset of G by A5; A10: now let o be Element of O; let a be Element of G; assume A11: a in M; now let Y be set; assume A12: Y in X; then consider H be strict StableSubgroup of G such that A13: Y = carr H and A c= Y by A1; a in carr H by A11,A12,A13,SETFAM_1:def 1; then a in H by STRUCT_0:def 5; then (G^o).a in H by Lm10; hence (G^o).a in Y by A13,STRUCT_0:def 5; end; hence (G^o).a in M by A3,SETFAM_1:def 1; end; A14: now let a,b be Element of G; assume that A15: a in M and A16: b in M; now let Y be set; assume A17: Y in X; then consider H be strict StableSubgroup of G such that A18: Y = carr H and A c= Y by A1; b in carr H by A16,A17,A18,SETFAM_1:def 1; then A19: b in H by STRUCT_0:def 5; a in carr H by A15,A17,A18,SETFAM_1:def 1; then a in H by STRUCT_0:def 5; then a * b in H by A19,Lm19; hence a * b in Y by A18,STRUCT_0:def 5; end; hence a * b in M by A3,SETFAM_1:def 1; end; now let a be Element of G; assume A20: a in M; now let Y be set; assume A21: Y in X; then consider H be strict StableSubgroup of G such that A22: Y = carr H and A c= Y by A1; a in carr H by A20,A21,A22,SETFAM_1:def 1; then a in H by STRUCT_0:def 5; then a" in H by Lm20; hence a" in Y by A22,STRUCT_0:def 5; end; hence a" in M by A3,SETFAM_1:def 1; end; then consider H be strict StableSubgroup of G such that A23: the carrier of H = M by A9,A14,A10,Lm15; take H; now let Y be set; assume Y in X; then ex H being strict StableSubgroup of G st Y = carr H & A c= Y by A1; hence A c= Y; end; hence A c= the carrier of H by A3,A23,SETFAM_1:5; let H1 be strict StableSubgroup of G; A24: the carrier of H1 = carr H1; assume A c= the carrier of H1; then the carrier of H1 in X by A1,A24; hence thesis by A23,Lm21,SETFAM_1:3; end; uniqueness proof let H1,H2 be strict StableSubgroup of G; assume that A25: A c= the carrier of H1 and A26: ( for H being strict StableSubgroup of G st A c= the carrier of H holds H1 is StableSubgroup of H)& A c= the carrier of H2 and A27: for H being strict StableSubgroup of G st A c= the carrier of H holds H2 is StableSubgroup of H; H1 is StableSubgroup of H2 by A26; then H1 is Subgroup of H2 by Def7; then A28: the carrier of H1 c= the carrier of H2 by GROUP_2:def 5; H2 is StableSubgroup of H1 by A25,A27; then H2 is Subgroup of H1 by Def7; then the carrier of H2 c= the carrier of H1 by GROUP_2:def 5; then the carrier of H1 = the carrier of H2 by A28,XBOOLE_0:def 10; hence thesis by Lm5; end; end; definition let O be set; let G be GroupWithOperators of O; let H1,H2 be StableSubgroup of G; func H1 "\/" H2 -> strict StableSubgroup of G equals the_stable_subgroup_of (carr H1 \/ carr H2); correctness; end; begin :: Some Theorems on Groups reformulated for Groups with Operators reserve x,O for set, o for Element of O, G,H,I for GroupWithOperators of O, A, B for Subset of G, N for normal StableSubgroup of G, H1,H2,H3 for StableSubgroup of G, g1,g2 for Element of G, h1,h2 for Element of H1, h for Homomorphism of G,H; :: GROUP_2:49 theorem Th1: x in H1 implies x in G proof assume A1: x in H1; H1 is Subgroup of G by Def7; hence thesis by A1,GROUP_2:40; end; :: GROUP_2:51 theorem Th2: h1 is Element of G proof H1 is Subgroup of G by Def7; hence thesis by GROUP_2:42; end; :: GROUP_2:52 theorem Th3: h1 = g1 & h2 = g2 implies h1 * h2 = g1 * g2 proof assume A1: h1 = g1 & h2 = g2; H1 is Subgroup of G by Def7; hence thesis by A1,GROUP_2:43; end; :: GROUP_2:53 theorem Th4: 1_G = 1_H1 proof reconsider H19=H1 as Subgroup of G by Def7; 1_H1 = 1_H19; hence thesis by GROUP_2:44; end; :: GROUP_2:55 theorem 1_G in H1 by Lm18; :: GROUP_2:57 theorem Th6: h1 = g1 implies h1" = g1" proof reconsider g9 = h1" as Element of G by Th2; A1: h1 * h1" = 1_H1 by GROUP_1:def 5; assume h1 = g1; then g1 * g9 = 1_H1 by A1,Th3 .= 1_G by Th4; hence thesis by GROUP_1:12; end; :: GROUP_2:59 theorem g1 in H1 & g2 in H1 implies g1 * g2 in H1 by Lm19; :: GROUP_2:60 theorem g1 in H1 implies g1" in H1 by Lm20; :: GROUP_2:61 theorem A <> {} & (for g1,g2 st g1 in A & g2 in A holds g1 * g2 in A) & (for g1 st g1 in A holds g1" in A) & (for o,g1 st g1 in A holds (G^o).g1 in A) implies ex H being strict StableSubgroup of G st the carrier of H = A by Lm15; :: GROUP_2:63 theorem Th10: G is StableSubgroup of G proof A1: now let o be Element of O; thus G^o = (G^o)|the carrier of G; end; G is Subgroup of G by GROUP_2:54; hence thesis by A1,Def7; end; :: GROUP_2:65 theorem Th11: for G1,G2,G3 being GroupWithOperators of O holds G1 is StableSubgroup of G2 & G2 is StableSubgroup of G3 implies G1 is StableSubgroup of G3 proof let G1,G2,G3 be GroupWithOperators of O; assume that A1: G1 is StableSubgroup of G2 and A2: G2 is StableSubgroup of G3; A3: G1 is Subgroup of G2 by A1,Def7; A4: now let o be Element of O; A5: the carrier of G1 c= the carrier of G2 by A3,GROUP_2:def 5; G1^o = (G2^o)|the carrier of G1 by A1,Def7 .= ((G3^o)|the carrier of G2)|the carrier of G1 by A2,Def7 .= (G3^o)|((the carrier of G2) /\ the carrier of G1) by RELAT_1:71; hence G1^o = (G3^o)|the carrier of G1 by A5,XBOOLE_1:28; end; G2 is Subgroup of G3 by A2,Def7; then G1 is Subgroup of G3 by A3,GROUP_2:56; hence thesis by A4,Def7; end; :: GROUP_2:66 theorem the carrier of H1 c= the carrier of H2 implies H1 is StableSubgroup of H2 by Lm21; :: GROUP_2:67 theorem Th13: (for g being Element of G st g in H1 holds g in H2) implies H1 is StableSubgroup of H2 proof assume A1: for g being Element of G st g in H1 holds g in H2; the carrier of H1 c= the carrier of H2 proof let x; assume x in the carrier of H1; then reconsider g = x as Element of H1; reconsider g as Element of G by Th2; g in H1 by STRUCT_0:def 5; then g in H2 by A1; hence thesis by STRUCT_0:def 5; end; hence thesis by Lm21; end; :: GROUP_2:68 theorem for H1,H2 being strict StableSubgroup of G st the carrier of H1 = the carrier of H2 holds H1 = H2 by Lm5; :: GROUP_2:75 theorem Th15: (1).G = (1).H1 proof A1: 1_H1 = 1_G by Th4; (1).H1 is StableSubgroup of G & the carrier of (1).H1 = {1_H1} by Def8,Th11; hence thesis by A1,Def8; end; :: GROUP_2:77 theorem Th16: (1).G is StableSubgroup of H1 proof (1).G = (1).H1 by Th15; hence thesis; end; :: GROUP_2:93 theorem Th17: carr H1 * carr H2 = carr H2 * carr H1 implies ex H being strict StableSubgroup of G st the carrier of H=carr H1 * carr H2 proof assume A1: carr H1 * carr H2 = carr H2 * carr H1; A2: now let o be Element of O; let g be Element of G; assume g in carr H1 * carr H2; then consider a,b be Element of G such that A3: g = a * b and A4: a in carr H1 and A5: b in carr H2; a in H1 by A4,STRUCT_0:def 5; then (G^o).a in H1 by Lm10; then A6: (G^o).a in carr H1 by STRUCT_0:def 5; b in H2 by A5,STRUCT_0:def 5; then (G^o).b in H2 by Lm10; then (G^o).b in carr H2 by STRUCT_0:def 5; then ((G^o).a) * ((G^o).b) in carr H1 * carr H2 by A6; hence (G^o).g in carr H1 * carr H2 by A3,GROUP_6:def 6; end; A7: H2 is Subgroup of G by Def7; A8: H1 is Subgroup of G by Def7; A9: now let g be Element of G; assume A10: g in carr H1 * carr H2; then consider a,b be Element of G such that A11: g = a * b and a in carr H1 and b in carr H2; consider b1,a1 be Element of G such that A12: a * b = b1 * a1 and A13: b1 in carr H2 and A14: a1 in carr H1 by A1,A10,A11; b1 in H2 by A13,STRUCT_0:def 5; then b1" in H2 by A7,GROUP_2:51; then A15: b1" in carr H2 by STRUCT_0:def 5; a1 in H1 by A14,STRUCT_0:def 5; then a1" in H1 by A8,GROUP_2:51; then A16: a1" in carr H1 by STRUCT_0:def 5; g" = a1" * b1" by A11,A12,GROUP_1:17; hence g" in carr H1 * carr H2 by A16,A15; end; A17: now let g1,g2 be Element of G; assume that A18: g1 in carr(H1) * carr(H2) and A19: g2 in carr(H1) * carr(H2); consider a1,b1 be Element of G such that A20: g1 = a1 * b1 and A21: a1 in carr(H1) and A22: b1 in carr(H2) by A18; consider a2,b2 be Element of G such that A23: g2 = a2 * b2 and A24: a2 in carr H1 and A25: b2 in carr H2 by A19; b1 * a2 in carr H1 * carr H2 by A1,A22,A24; then consider a,b be Element of G such that A26: b1 * a2 = a * b and A27: a in carr H1 and A28: b in carr H2; A29: a in H1 by A27,STRUCT_0:def 5; A30: b in H2 by A28,STRUCT_0:def 5; b2 in H2 by A25,STRUCT_0:def 5; then b * b2 in H2 by A7,A30,GROUP_2:50; then A31: b * b2 in carr H2 by STRUCT_0:def 5; a1 in H1 by A21,STRUCT_0:def 5; then a1 * a in H1 by A8,A29,GROUP_2:50; then A32: a1 * a in carr H1 by STRUCT_0:def 5; g1 * g2 = a1 * b1 * a2 * b2 by A20,A23,GROUP_1:def 3 .= a1 * (b1 * a2) * b2 by GROUP_1:def 3; then g1 * g2 = a1 * a * b * b2 by A26,GROUP_1:def 3 .= a1 * a * (b * b2) by GROUP_1:def 3; hence g1 * g2 in carr H1 * carr H2 by A32,A31; end; carr H1 * carr H2 <> {} by GROUP_2:9; hence thesis by A17,A9,A2,Lm15; end; :: GROUP_2:97 theorem Th18: (for H being StableSubgroup of G st H = H1 /\ H2 holds the carrier of H = (the carrier of H1) /\ (the carrier of H2)) & for H being strict StableSubgroup of G holds the carrier of H = (the carrier of H1) /\ (the carrier of H2) implies H = H1 /\ H2 proof A1: the carrier of H1 = carr(H1) & the carrier of H2 = carr(H2); thus for H being StableSubgroup of G st H = H1 /\ H2 holds the carrier of H = (the carrier of H1) /\ (the carrier of H2) proof let H be StableSubgroup of G; assume H = H1 /\ H2; hence the carrier of H = carr(H1)/\carr(H2) by Def25 .= (the carrier of H1)/\(the carrier of H2); end; let H be strict StableSubgroup of G; assume the carrier of H = (the carrier of H1) /\ (the carrier of H2); hence thesis by A1,Def25; end; :: GROUP_2:100 theorem Th19: for H being strict StableSubgroup of G holds H /\ H = H proof let H be strict StableSubgroup of G; the carrier of H /\ H = carr(H) /\ carr(H) by Def25 .= the carrier of H; hence thesis by Lm5; end; :: GROUP_2:102 theorem Th20: H1 /\ H2 /\ H3 = H1 /\ (H2 /\ H3) proof the carrier of H1 /\ H2 /\ H3 = carr(H1 /\ H2) /\ carr(H3) by Def25 .= carr(H1) /\ carr(H2) /\ carr(H3) by Def25 .= carr(H1) /\ (carr(H2) /\ carr(H3)) by XBOOLE_1:16 .= carr(H1) /\ carr(H2 /\ H3) by Def25 .= the carrier of H1 /\ (H2 /\ H3) by Def25; hence thesis by Lm5; end; Lm22: for H1 being strict StableSubgroup of G holds H1 is StableSubgroup of H2 iff H1 /\ H2 = H1 proof let H1 be strict StableSubgroup of G; thus H1 is StableSubgroup of H2 implies H1 /\ H2 = H1 proof assume H1 is StableSubgroup of H2; then H1 is Subgroup of H2 by Def7; then A1: the carrier of H1 c= the carrier of H2 by GROUP_2:def 5; the carrier of H1 /\ H2 = carr(H1) /\ carr (H2) by Def25; hence thesis by A1,Lm5,XBOOLE_1:28; end; assume H1 /\ H2 = H1; then the carrier of H1 = carr(H1) /\ carr(H2) by Def25 .= (the carrier of H1) /\ (the carrier of H2); hence thesis by Lm21,XBOOLE_1:17; end; :: GROUP_2:103 theorem Th21: (1).G /\ H1 = (1).G & H1 /\ (1).G = (1).G proof A1: (1).G is StableSubgroup of H1 by Th16; hence (1).G /\ H1 = (1).G by Lm22; thus thesis by A1,Lm22; end; :: GROUP_2:167 theorem Th22: union Cosets N = the carrier of G proof reconsider H = the multMagma of N as strict normal Subgroup of G by Lm7; now set h = the Element of H; let x; reconsider g = h as Element of G by GROUP_2:42; assume x in the carrier of G; then reconsider a = x as Element of G; A1: a = a * 1_G by GROUP_1:def 4 .= a * (g" * g) by GROUP_1:def 5 .= a * g" * g by GROUP_1:def 3; A2: a * g" * H in Cosets H by GROUP_2:def 15; h in H by STRUCT_0:def 5; then a in a * g" * H by A1,GROUP_2:103; hence x in union Cosets H by A2,TARSKI:def 4; end; then A3: the carrier of G c= union Cosets H by TARSKI:def 3; Cosets N = Cosets H by Def14; hence thesis by A3,XBOOLE_0:def 10; end; :: GROUP_3:149 theorem Th23: for N1,N2 being strict normal StableSubgroup of G ex N being strict normal StableSubgroup of G st the carrier of N = carr N1 * carr N2 proof let N1,N2 be strict normal StableSubgroup of G; set N19 = the multMagma of N1; set N29 = the multMagma of N2; reconsider N19,N29 as strict normal Subgroup of G by Lm7; set A = carr N19 * carr N29; set B = carr N19; set C = carr N29; carr N19 * carr N29 = carr N29 * carr N19 by GROUP_3:125; then consider H9 be strict Subgroup of G such that A1: the carrier of H9 = A by GROUP_2:78; A2: now let o be Element of O; let g be Element of G; assume g in A; then consider a,b be Element of G such that A3: g = a * b and A4: a in carr N1 and A5: b in carr N2; a in N1 by A4,STRUCT_0:def 5; then (G^o).a in N1 by Lm10; then A6: (G^o).a in carr N1 by STRUCT_0:def 5; b in N2 by A5,STRUCT_0:def 5; then (G^o).b in N2 by Lm10; then (G^o).b in carr N2 by STRUCT_0:def 5; then ((G^o).a) * ((G^o).b) in carr N1 * carr N2 by A6; hence (G^o).g in A by A3,GROUP_6:def 6; end; A7: now let g be Element of G; assume g in A; then g in H9 by A1,STRUCT_0:def 5; then g" in H9 by GROUP_2:51; hence g" in A by A1,STRUCT_0:def 5; end; now let g1,g2 be Element of G; assume g1 in A & g2 in A; then g1 in H9 & g2 in H9 by A1,STRUCT_0:def 5; then g1 * g2 in H9 by GROUP_2:50; hence g1 * g2 in A by A1,STRUCT_0:def 5; end; then consider H be strict StableSubgroup of G such that A8: the carrier of H = A by A1,A7,A2,Lm15; now let a be Element of G; thus a * H9 = a * N19 * C by A1,GROUP_2:29 .= N19 * a * C by GROUP_3:117 .= B * (a * N29) by GROUP_2:30 .= B * (N29 * a) by GROUP_3:117 .= H9 * a by A1,GROUP_2:31; end; then H9 is normal Subgroup of G by GROUP_3:117; then for H99 being strict Subgroup of G st H99 = the multMagma of H holds H99 is normal by A1,A8,GROUP_2:59; then H is normal by Def10; hence thesis by A8; end; Lm23: for F1 being FinSequence, y being Element of NAT st y in dom F1 holds len F1 - y + 1 is Element of NAT & len F1 - y + 1 >= 1 & len F1 - y + 1 <= len F1 proof let F1 be FinSequence, y be Element of NAT; assume A1: y in dom F1; now assume len F1 - y + 1 < 0; then 1 < 0 qua Nat - (len F1 - y) by XREAL_1:20; then 1 < y - len F1; then A2: len F1 + 1 < y by XREAL_1:20; y <= len F1 by A1,FINSEQ_3:25; hence contradiction by A2,NAT_1:12; end; then reconsider n = len F1 - y + 1 as Element of NAT by INT_1:3; y >= 1 by A1,FINSEQ_3:25; then n - 1 - y <= len F1 - y - 1 by XREAL_1:13; then A3: n - (y + 1) <= len F1 - (y + 1); y + 0 <= len F1 by A1,FINSEQ_3:25; then 0 + 1 = 1 & 0 <= len F1 - y by XREAL_1:19; hence thesis by A3,XREAL_1:6,9; end; Lm24: for G,H being Group, F1 being FinSequence of the carrier of G, F2 being FinSequence of the carrier of H, I being FinSequence of INT, f being Homomorphism of G,H st (for k being Nat st k in dom F1 holds F2.k = f.(F1.k)) & len F1 = len I & len F2 = len I holds f.(Product(F1 |^ I)) = Product(F2 |^ I) proof defpred P[Nat] means for G,H being Group, F1 being FinSequence of the carrier of G, F2 being FinSequence of the carrier of H, I being FinSequence of INT, f being Homomorphism of G,H st (for k being Nat st k in dom F1 holds F2.k = f.(F1.k)) & len F1 = len I & len F2 = len I & $1 = len I holds f.(Product(F1 |^ I)) = Product(F2 |^ I); let G,H be Group; let F1 be FinSequence of the carrier of G; let F2 be FinSequence of the carrier of H; let I be FinSequence of INT; let f be Homomorphism of G,H; assume A1: ( for k being Nat st k in dom F1 holds F2.k = f.(F1.k))& len F1 = len I & len F2 = len I; A2: now let n be Nat; assume A3: P[n]; thus P[n+1] proof let G,H be Group; let F1 be FinSequence of the carrier of G; let F2 be FinSequence of the carrier of H; let I be FinSequence of INT; let f be Homomorphism of G,H; assume A4: for k being Nat st k in dom F1 holds F2.k = f.(F1.k); assume that A5: len F1 = len I and A6: len F2 = len I and A7: n+1 = len I; consider F1n be FinSequence of the carrier of G, g be Element of G such that A8: F1 = F1n^<*g*> by A5,A7,FINSEQ_2:19; A9: len F1 = len F1n + len <*g*> by A8,FINSEQ_1:22; then A10: n+1 = len F1n + 1 by A5,A7,FINSEQ_1:40; consider F2n be FinSequence of the carrier of H, h be Element of H such that A11: F2 = F2n^<*h*> by A6,A7,FINSEQ_2:19; A12: dom F1 = dom F2 & dom F2 = dom I by A5,A6,FINSEQ_3:29; 1 <= n+1 by NAT_1:11; then A13: n+1 in dom I by A7,FINSEQ_3:25; set F21=<*h*>; set F11=<*g*>; consider In be FinSequence of INT, i be Element of INT such that A14: I = In^<*i*> by A7,FINSEQ_2:19; set I1=<*i*>; len I = len In + len <*i*> by A14,FINSEQ_1:22; then A15: n+1 = len In + 1 by A7,FINSEQ_1:40; A16: len F2 = len F2n + len <*h*> by A11,FINSEQ_1:22; then A17: n+1 = len F2n + 1 by A6,A7,FINSEQ_1:40; A18: now let k be Nat; 0+n <= 1+n by XREAL_1:6; then A19: dom F1n c= dom F1 by A5,A7,A10,FINSEQ_3:30; assume A20: k in dom F1n; then k in dom F2n by A10,A17,FINSEQ_3:29; hence F2n.k = F2.k by A11,FINSEQ_1:def 7 .= f.(F1.k) by A4,A20,A19 .= f.(F1n.k) by A8,A20,FINSEQ_1:def 7; end; A21: F2.(n+1)=(F2n^<*h*>).(len F2n +1) by A6,A7,A11,A16,FINSEQ_1:40 .= h by FINSEQ_1:42; A22: F1.(n+1)=(F1n^<*g*>).(len F1n +1) by A5,A7,A8,A9,FINSEQ_1:40 .= g by FINSEQ_1:42; len F21 = 1 by FINSEQ_1:40 .=len I1 by FINSEQ_1:40; then A23: Product(F2 |^ I) = Product((F2n |^ In)^(F21 |^ I1)) by A14,A11,A15,A17, GROUP_4:19 .= Product(F2n |^ In) * Product(F21 |^ I1) by GROUP_4:5; A24: f.Product(F11 |^ I1) = f.Product(<*g*>|^<*@i*>) .= f.Product <*g|^i*> by GROUP_4:22 .= f.(g|^i) by GROUP_4:9 .= (f.g)|^i by GROUP_6:37 .= h|^i by A4,A13,A12,A22,A21 .= Product <*h|^i*> by GROUP_4:9 .= Product(<*h*>|^<*@i*>) by GROUP_4:22 .= Product(F21 |^ I1); len F11 = 1 by FINSEQ_1:40 .=len I1 by FINSEQ_1:40; then Product(F1 |^ I) = Product((F1n |^ In)^(F11 |^ I1)) by A14,A8,A15,A10, GROUP_4:19 .= Product(F1n |^ In) * Product(F11 |^ I1) by GROUP_4:5; then f.(Product(F1 |^ I)) = f.Product(F1n |^ In) * f.Product(F11 |^ I1) by GROUP_6:def 6 .= Product(F2n |^ In) * Product(F21 |^ I1) by A3,A15,A10,A17,A18,A24; hence thesis by A23; end; end; A25: P[0] proof let G,H be Group; let F1 be FinSequence of the carrier of G; let F2 be FinSequence of the carrier of H; let I be FinSequence of INT; let f be Homomorphism of G,H; assume for k being Nat st k in dom F1 holds F2.k = f.(F1.k); assume that A26: len F1 = len I and A27: len F2 = len I and A28: 0 = len I; len(F2 |^ I) = 0 by A27,A28,GROUP_4:def 3; then F2 |^ I = <*> the carrier of H; then A29: Product(F2 |^ I) = 1_H by GROUP_4:8; len(F1 |^ I) = 0 by A26,A28,GROUP_4:def 3; then F1 |^ I = <*> the carrier of G; then Product(F1 |^ I) = 1_G by GROUP_4:8; hence thesis by A29,GROUP_6:31; end; for n being Nat holds P[n] from NAT_1:sch 2(A25,A2); hence thesis by A1; end; :: GROUP_4:37 theorem Th24: g1 in the_stable_subgroup_of A iff ex F being FinSequence of the carrier of G, I being FinSequence of INT, C being Subset of G st C = the_stable_subset_generated_by (A, the action of G) & len F = len I & rng F c= C & Product(F |^ I) = g1 proof set H9 = the_stable_subgroup_of A; set Y = the carrier of H9; A1: A c= the carrier of H9 by Def26; thus g1 in the_stable_subgroup_of A implies ex F being FinSequence of the carrier of G, I being FinSequence of INT, C being Subset of G st C = the_stable_subset_generated_by (A, the action of G) & len F = len I & rng F c= C & Product(F |^ I) = g1 proof defpred P[set] means ex F being FinSequence of the carrier of G, I being FinSequence of INT, C being Subset of G st C = the_stable_subset_generated_by ( A, the action of G) & $1 = Product (F |^ I) & len F = len I & rng F c= C; assume A2: g1 in the_stable_subgroup_of A; reconsider B = {b where b is Element of G : P[b]} as Subset of G from DOMAIN_1:sch 7; A3: now let c,d be Element of G; assume that A4: c in B and A5: d in B; ex d1 being Element of G st c = d1 & ex F being FinSequence of the carrier of G, I being FinSequence of INT, C being Subset of G st C = the_stable_subset_generated_by (A, the action of G) & d1 = Product(F |^ I) & len F = len I & rng F c= C by A4; then consider F1 be FinSequence of the carrier of G, I1 be FinSequence of INT , C be Subset of G such that A6: C = the_stable_subset_generated_by (A, the action of G) and A7: c = Product(F1 |^ I1) and A8: len F1 = len I1 and A9: rng F1 c= C; ex d2 being Element of G st d = d2 & ex F being FinSequence of the carrier of G, I being FinSequence of INT, C being Subset of G st C = the_stable_subset_generated_by (A, the action of G) & d2 = Product(F |^ I) & len F = len I & rng F c= C by A5; then consider F2 be FinSequence of the carrier of G, I2 be FinSequence of INT , C be Subset of G such that A10: C = the_stable_subset_generated_by (A, the action of G) and A11: d = Product(F2 |^ I2) and A12: len F2 = len I2 and A13: rng F2 c= C; A14: len(F1 ^ F2) = len I1 + len I2 by A8,A12,FINSEQ_1:22 .= len(I1 ^ I2) by FINSEQ_1:22; rng(F1 ^ F2) = rng F1 \/ rng F2 by FINSEQ_1:31; then A15: rng(F1 ^ F2) c= C by A6,A9,A10,A13,XBOOLE_1:8; c * d = Product((F1 |^ I1) ^ (F2 |^ I2)) by A7,A11,GROUP_4:5 .= Product((F1 ^ F2) |^ (I1 ^ I2)) by A8,A12,GROUP_4:19; hence c * d in B by A10,A15,A14; end; A16: now let o be Element of O; let c be Element of G; assume c in B; then ex d1 being Element of G st c = d1 & ex F being FinSequence of the carrier of G, I being FinSequence of INT, C being Subset of G st C = the_stable_subset_generated_by (A, the action of G) & d1 = Product(F |^ I) & len F = len I & rng F c= C; then consider F1 be FinSequence of the carrier of G, I1 be FinSequence of INT , C be Subset of G such that A17: C = the_stable_subset_generated_by (A, the action of G) and A18: c = Product(F1 |^ I1) and A19: len F1 = len I1 and A20: rng F1 c= C; deffunc F(Nat) = (G^o).(F1.$1); consider F2 being FinSequence such that A21: len F2 = len F1 and A22: for k being Nat st k in dom F2 holds F2.k = F(k) from FINSEQ_1: sch 2; A23: dom F2 = dom F1 by A21,FINSEQ_3:29; A24: Seg len F1 = dom F1 by FINSEQ_1:def 3; now A25: C is_stable_under_the_action_of the action of G by A17,Def2; let y be set; assume y in rng F2; then consider x such that A26: x in dom F2 and A27: y = F2.x by FUNCT_1:def 3; A28: x in Seg len F1 by A21,A26,FINSEQ_1:def 3; reconsider x as Element of NAT by A26; A29: F2.x = (G^o).(F1.x) by A22,A26; A30: F1.x in rng F1 by A24,A28,FUNCT_1:3; per cases; suppose A31: O<>{}; set f = (the action of G).o; A32: G^o = (the action of G).o by A31,Def6; then reconsider f as Function of G, G; dom f = the carrier of G by FUNCT_2:def 1; then A33: y in f .: C by A20,A27,A29,A30,A32,FUNCT_1:def 6; (f .: C) c= C by A25,A31,Def1; hence y in C by A33; end; suppose O={}; then G^o = id the carrier of G by Def6; then (G^o).(F1.x) = F1.x by A30,FUNCT_1:18; hence y in C by A20,A27,A29,A30; end; end; then A34: rng F2 c= C by TARSKI:def 3; then rng F2 c= the carrier of G by XBOOLE_1:1; then reconsider F2 as FinSequence of the carrier of G by FINSEQ_1:def 4; (G^o).c = Product(F2 |^ I1) by A18,A19,A21,A22,A23,Lm24; hence (G^o).c in B by A17,A19,A21,A34; end; A35: now let c be Element of G; assume c in B; then ex d1 being Element of G st c = d1 & ex F being FinSequence of the carrier of G, I being FinSequence of INT, C being Subset of G st C = the_stable_subset_generated_by (A, the action of G) & d1 = Product(F |^ I) & len F = len I & rng F c= C; then consider F1 be FinSequence of the carrier of G, I1 be FinSequence of INT , C be Subset of G such that A36: C = the_stable_subset_generated_by (A, the action of G) & c = Product(F1 |^ I1) and A37: len F1 = len I1 and A38: rng F1 c= C; deffunc F(Nat) = F1.(len F1 - $1 + 1); consider F2 being FinSequence such that A39: len F2 = len F1 and A40: for k being Nat st k in dom F2 holds F2.k = F(k) from FINSEQ_1: sch 2; A41: Seg len I1 = dom I1 by FINSEQ_1:def 3; A42: rng F2 c= rng F1 proof let x; assume x in rng F2; then consider y be set such that A43: y in dom F2 and A44: F2.y = x by FUNCT_1:def 3; reconsider y as Element of NAT by A43; reconsider n = len F1 - y + 1 as Element of NAT by A39,A43,Lm23; 1 <= n & n <= len F1 by A39,A43,Lm23; then A45: n in dom F1 by FINSEQ_3:25; x = F1.(len F1 - y + 1) by A40,A43,A44; hence thesis by A45,FUNCT_1:def 3; end; then A46: rng F2 c= C by A38,XBOOLE_1:1; set p = F1 |^ I1; A47: Seg len F1 = dom F1 by FINSEQ_1:def 3; A48: len p = len F1 by GROUP_4:def 3; defpred P[Nat,set] means ex i being Integer st i = I1.(len I1 - $1 + 1) & $2 = - i; A49: for k being Nat st k in Seg len I1 ex x st P[k,x] proof let k be Nat; assume k in Seg len I1; then A50: k in dom I1 by FINSEQ_1:def 3; then reconsider n = len I1 - k + 1 as Element of NAT by Lm23; 1 <= n & n <= len I1 by A50,Lm23; then n in dom I1 by FINSEQ_3:25; then I1.n in rng I1 by FUNCT_1:def 3; then reconsider i = I1.n as Element of INT qua non empty set; reconsider i as Integer; reconsider x = - i as set; take x,i; thus thesis; end; consider I2 being FinSequence such that A51: dom I2 = Seg len I1 and A52: for k be Nat st k in Seg len I1 holds P[k,I2.k] from FINSEQ_1: sch 1(A49); A53: len F2 = len I2 by A37,A39,A51,FINSEQ_1:def 3; A54: rng I2 c= INT proof let x; assume x in rng I2; then consider y be set such that A55: y in dom I2 and A56: x = I2.y by FUNCT_1:def 3; reconsider y as Element of NAT by A55; ex i being Integer st i = I1.(len I1 - y + 1) & x = - i by A51,A52,A55 ,A56; hence thesis by INT_1:def 2; end; A57: rng F2 c= the carrier of G by A42,XBOOLE_1:1; A58: dom F2 = dom I2 by A37,A39,A51,FINSEQ_1:def 3; reconsider I2 as FinSequence of INT by A54,FINSEQ_1:def 4; reconsider F2 as FinSequence of the carrier of G by A57,FINSEQ_1:def 4; set q = F2 |^ I2; A59: len q = len F2 by GROUP_4:def 3; then A60: dom q = dom F2 by FINSEQ_3:29; A61: dom F1 = dom I1 by A37,FINSEQ_3:29; now let k be Element of NAT; A62: I2/.k = @(I2/.k); assume A63: k in dom q; then reconsider n = len p - k + 1 as Element of NAT by A39,A48,A59,Lm23 ; A64: I1/.n = @(I1/.n) & q/.k = q.k by A63,PARTFUN1:def 6; A65: F2/.k = F2.k & F2.k = F1.n by A40,A48,A60,A63,PARTFUN1:def 6; 1 <= n & len p >= n by A39,A48,A59,A63,Lm23; then A66: n in dom I2 by A37,A51,A48; then A67: I1.n = I1/.n by A51,A41,PARTFUN1:def 6; dom q = dom I1 by A37,A39,A59,FINSEQ_3:29; then consider i be Integer such that A68: i = I1.n and A69: I2.k = - i by A37,A52,A41,A48,A63; I2.k = I2/.k by A58,A60,A63,PARTFUN1:def 6; then A70: q.k = (F2/.k) |^ (- i) by A60,A63,A69,A62,GROUP_4:def 3; F1/.n= F1.n by A37,A47,A51,A66,PARTFUN1:def 6; then q.k = ((F1/.n) |^ i)" by A70,A65,GROUP_1:36; hence (q/.k)" = p.(len p - k + 1) by A61,A51,A41,A66,A68,A67,A64, GROUP_4:def 3; end; then Product p" = Product q by A39,A48,A59,GROUP_4:14; hence c" in B by A36,A53,A46; end; A71: len {} = 0; A72: rng <*> the carrier of G = {} & {} c= the_stable_subset_generated_by ( A, the action of G) by XBOOLE_1:2; 1_G = Product <*> the carrier of G & (<*> the carrier of G) |^ (<*> INT) = {} by GROUP_4:8,21; then 1_G in B by A72,A71; then consider H be strict StableSubgroup of G such that A73: the carrier of H = B by A3,A35,A16,Lm15; A c= B proof set C = the_stable_subset_generated_by (A, the action of G); reconsider p = 1 as Integer; let x; assume A74: x in A; then reconsider a = x as Element of G; A c= C by Def2; then A75: rng<* a *> = {a} & {a} c= C by A74,FINSEQ_1:39,ZFMISC_1:31; A76: Product(<* a *> |^ <* @p *>) = Product<* a |^ 1 *> by GROUP_4:22 .= a |^ 1 by GROUP_4:9 .= a by GROUP_1:26; len<* a *> = 1 & len<* @p *> = 1 by FINSEQ_1:39; hence thesis by A76,A75; end; then the_stable_subgroup_of A is StableSubgroup of H by A73,Def26; then the_stable_subgroup_of A is Subgroup of H by Def7; then g1 in H by A2,GROUP_2:40; then g1 in B by A73,STRUCT_0:def 5; then ex b being Element of G st b = g1 & ex F being FinSequence of the carrier of G, I being FinSequence of INT, C being Subset of G st C = the_stable_subset_generated_by (A, the action of G) & b = Product(F |^ I) & len F = len I & rng F c= C; hence thesis; end; given F be FinSequence of the carrier of G, I be FinSequence of INT, C be Subset of G such that A77: C = the_stable_subset_generated_by (A, the action of G) and len F = len I and A78: rng F c= C and A79: Product(F |^ I) = g1; H9 is Subgroup of G by Def7; then reconsider Y as Subset of G by GROUP_2:def 5; now let o be Element of O; let f be Function of G, G; assume A80: o in O; assume A81: f = (the action of G).o; now let y be set; assume y in f .: Y; then consider x such that A82: x in dom f and A83: x in Y and A84: y = f.x by FUNCT_1:def 6; reconsider x as Element of G by A82; x in H9 by A83,STRUCT_0:def 5; then (G^o).x in H9 by Lm10; then f.x in H9 by A80,A81,Def6; hence y in Y by A84,STRUCT_0:def 5; end; hence (f .: Y) c= Y by TARSKI:def 3; end; then A85: Y is_stable_under_the_action_of the action of G by Def1; reconsider H9 as Subgroup of G by Def7; C c= the carrier of H9 by A77,A1,A85,Def2; then rng F c= carr H9 by A78,XBOOLE_1:1; hence thesis by A79,GROUP_4:20; end; Lm25: A is empty implies the_stable_subgroup_of A = (1).G proof A1: now let H be strict StableSubgroup of G; assume A c= the carrier of H; (1).G = (1).H by Th15; hence (1).G is StableSubgroup of H; end; assume A is empty; then A c= the carrier of (1).G by XBOOLE_1:2; hence thesis by A1,Def26; end; Lm26: for O being non empty set, E being set, o being Element of O, A being Action of O,E holds Product(<*o*>,A) = A.o proof let O be non empty set; let E be set; let o be Element of O; let A be Action of O,E; len <*o*> = 1 & ex PF be FinSequence of Funcs(E,E) st Product(<*o*>,A) = PF .(len <*o*>) & len PF = len <*o*> & PF.1 = A.(<*o*>.1) & for k being Nat st k<>0 & k holds ex f,g being Function of E,E st f = PF.k & g = A.( <*o *>.(k+1)) & PF.(k+1) = f*g by Def3,FINSEQ_1:39; hence thesis by FINSEQ_1:40; end; Lm27: for O being non empty set, E being set, o being Element of O, F being FinSequence of O, A being Action of O,E holds Product(F^<*o*>,A) = Product(F,A) *Product(<*o*>,A) proof let O be non empty set; let E be set; let o be Element of O; let F be FinSequence of O; let A be Action of O,E; set F1=F^<*o*>; A1: len F1 = len F + len <*o*> by FINSEQ_1:22 .= len F + 1 by FINSEQ_1:39; consider PF1 be FinSequence of Funcs(E,E) such that A2: Product(F1,A) = PF1.(len F1) and A3: len PF1 = len F1 and A4: PF1.1 = A.(F1.1) and A5: for k being Nat st k<>0 & k 0; reconsider PF=PF1|(Seg len F) as FinSequence of Funcs(E,E) by FINSEQ_1:18; set IT = PF.(len F); A7: Product(<*o*>,A) = A.o by Lm26 .= A.(F1.(len F + 1)) by FINSEQ_1:42; A8: now let k be Nat; assume A9: k<>0; then A10: 0+1O; hence Product(F^<*o*>,A) = Product(<*>O^<*o*>,A) .= Product(<*o*>,A) by FINSEQ_1:34 .= (id E)*Product(<*o*>,A) by FUNCT_2:17 .= Product(F,A)*Product(<*o*>,A) by A22,Def3; end; end; Lm28: for O being non empty set, E being set, o being Element of O, F being FinSequence of O, A being Action of O,E holds Product(<*o*>^F,A) = Product(<*o *>,A)*Product(F,A) proof let O be non empty set; let E be set; let o be Element of O; let F be FinSequence of O; let A be Action of O,E; defpred P[Element of NAT] means for F being FinSequence of O st len F = $1 holds Product(<*o*>^F,A) = Product(<*o*>,A)*Product(F,A); reconsider k = len F as Element of NAT; A1: k = len F; A2: for k being Element of NAT st P[k] holds P[k + 1] proof let k be Element of NAT; assume A3: P[k]; now let F be FinSequence of O; assume A4: len F = k+1; then consider Fk be FinSequence of O, o9 be Element of O such that A5: F = Fk^<*o9*> by FINSEQ_2:19; len F = len Fk + len <*o9*> by A5,FINSEQ_1:22; then A6: k+1 = len Fk + 1 by A4,FINSEQ_1:39; set F2k=<*o*>^Fk; thus Product(<*o*>^F,A) = Product(<*o*>^Fk^<*o9*>,A) by A5,FINSEQ_1:32 .= Product(F2k,A)*Product(<*o9*>,A) by Lm27 .= Product(<*o*>,A)*Product(Fk,A)*Product(<*o9*>,A) by A3,A6 .= Product(<*o*>,A)*((Product(Fk,A))*Product(<*o9*>,A)) by RELAT_1:36 .= Product(<*o*>,A)*Product(F,A) by A5,Lm27; end; hence thesis; end; A7: P[0] proof let F be FinSequence of O; assume A8: len F = 0; then F = <*>O; hence Product(<*o*>^F,A) = Product(<*o*>^<*>O,A) .= Product(<*o*>,A) by FINSEQ_1:34 .= Product(<*o*>,A)*id E by FUNCT_2:17 .= Product(<*o*>,A)*Product(F,A) by A8,Def3; end; for k being Element of NAT holds P[k] from NAT_1:sch 1(A7,A2); hence thesis by A1; end; Lm29: for O being non empty set, E being set, F1,F2 being FinSequence of O, A being Action of O,E holds Product(F1^F2,A) = Product(F1,A) * Product(F2,A) proof let O be non empty set, E be set; let F1,F2 be FinSequence of O; let A be Action of O,E; defpred P[Element of NAT] means for F1,F2 being FinSequence of O st len F1 = $1 holds Product(F1^F2,A) = Product(F1,A)*Product(F2,A); reconsider k = len F1 as Element of NAT; A1: k = len F1; A2: for k being Element of NAT st P[k] holds P[k + 1] proof let k be Element of NAT; assume A3: P[k]; now let F1,F2 be FinSequence of O; assume A4: len F1 = k+1; then consider F1k be FinSequence of O, o be Element of O such that A5: F1 = F1k^<*o*> by FINSEQ_2:19; set F2k=<*o*>^F2; len F1 = len F1k + len <*o*> by A5,FINSEQ_1:22; then A6: k+1 = len F1k + 1 by A4,FINSEQ_1:39; thus Product(F1^F2,A) = Product(F1k^F2k,A) by A5,FINSEQ_1:32 .= Product(F1k,A)*Product(F2k,A) by A3,A6 .= Product(F1k,A)*(Product(<*o*>,A)*Product(F2,A)) by Lm28 .= (Product(F1k,A)*Product(<*o*>,A))*Product(F2,A) by RELAT_1:36 .= Product(F1,A)*Product(F2,A) by A3,A5,A6; end; hence thesis; end; A7: P[0] proof let F1,F2 be FinSequence of O; assume A8: len F1 = 0; then F1 = <*>O; hence Product(F1^F2,A) = Product(<*>O^F2,A) .= Product(F2,A) by FINSEQ_1:34 .= (id E)*Product(F2,A) by FUNCT_2:17 .= Product(F1,A)*Product(F2,A) by A8,Def3; end; for k being Element of NAT holds P[k] from NAT_1:sch 1(A7,A2); hence thesis by A1; end; Lm30: for O,E being set, F being FinSequence of O, Y being Subset of E, A being Action of O,E st Y is_stable_under_the_action_of A holds Product(F,A) .: Y c= Y proof let O,E be set; let F be FinSequence of O; let Y be Subset of E; let A be Action of O,E; assume A1: Y is_stable_under_the_action_of A; per cases; suppose O={}; then len F = 0; then Product(F,A) = id E by Def3; hence thesis by FUNCT_1:92; end; suppose A2: O<>{}; defpred P[Element of NAT] means for F being FinSequence of O st len F = $1 holds Product(F,A) .: Y c= Y; A3: for k being Element of NAT st P[k] holds P[k + 1] proof let k be Element of NAT; assume A4: P[k]; now let F be FinSequence of O; assume A5: len F = k+1; then consider Fk be FinSequence of O, o be Element of O such that A6: F = Fk^<*o*> by FINSEQ_2:19; len F = len Fk + len <*o*> by A6,FINSEQ_1:22; then k+1 = len Fk + 1 by A5,FINSEQ_1:39; then A7: Product(Fk,A) .: Y c= Y by A4; reconsider F1 = <*o*> as FinSequence of O by A6,FINSEQ_1:36; Product(F,A) = Product(Fk,A)*Product(F1,A) by A2,A6,Lm29; then A8: Product(F,A) .: Y = Product(Fk,A) .: (Product(F1,A) .: Y) by RELAT_1:126; Product(F1,A) = A.o by A2,Lm26; then Product(F1,A) .: Y c= Y by A1,A2,Def1; then Product(F,A) .: Y c= Product(Fk,A) .: Y by A8,RELAT_1:123; hence Product(F,A) .: Y c= Y by A7,XBOOLE_1:1; end; hence thesis; end; reconsider k = len F as Element of NAT; A9: k = len F; A10: P[0] proof let F be FinSequence of O; assume len F = 0; then Product(F,A) = id E by Def3; hence thesis by FUNCT_1:92; end; for k being Element of NAT holds P[k] from NAT_1:sch 1(A10,A3); hence thesis by A9; end; end; Lm31: for E being non empty set, A being Action of O,E holds for X being Subset of E, a being Element of E st X is not empty holds a in the_stable_subset_generated_by(X,A) iff ex F being FinSequence of O, x being Element of X st Product(F,A).x = a proof let E be non empty set; let A be Action of O,E; let X be Subset of E; let a be Element of E; defpred P[set] means ex F being FinSequence of O, x being Element of X st Product(F,A).x = $1; set B = {e where e is Element of E: P[e]}; reconsider B as Subset of E from DOMAIN_1:sch 7; assume A1: X is not empty; A2: now let Y be Subset of E; assume A3: Y is_stable_under_the_action_of A; assume A4: X c= Y; now let x; assume x in B; then consider e be Element of E such that A5: x=e and A6: ex F being FinSequence of O, x9 being Element of X st Product(F ,A). x9 = e; consider F be FinSequence of O, x9 be Element of X such that A7: Product(F,A).x9 = e by A6; A8: x9 in X by A1; then x9 in E; then x9 in dom Product(F,A) by FUNCT_2:def 1; then A9: Product(F,A).x9 in Product(F,A).:Y by A4,A8,FUNCT_1:def 6; Product(F,A).:Y c= Y by A3,Lm30; hence x in Y by A5,A7,A9; end; hence B c= Y by TARSKI:def 3; end; now let o be Element of O; let f be Function of E, E; assume A10: o in O; assume A11: f = A.o; per cases; suppose O={}; hence (f .: B) c= B by A10; end; suppose A12: O<>{}; now reconsider o as Element of O; reconsider F99=<*o*> as FinSequence of O by A12,FINSEQ_1:74; let y be set; assume y in f .: B; then consider x such that A13: x in dom f and A14: x in B and A15: y = f.x by FUNCT_1:def 6; y in rng f by A13,A15,FUNCT_1:3; then reconsider e=y as Element of E; consider e9 be Element of E such that A16: e9=x and A17: ex F9 being FinSequence of O, x9 being Element of X st Product(F9,A ).x9 = e9 by A14; consider F9 be FinSequence of O, x9 be Element of X such that A18: Product(F9,A).x9 = e9 by A17; reconsider F=F99^F9 as FinSequence of O; x9 in X by A1; then x9 in E; then A19: x9 in dom Product(F9,A) by FUNCT_2:def 1; Product(F,A).x9 = (Product(F99,A)*Product(F9,A)).x9 by A12,Lm29 .= Product(F99,A).(Product(F9,A).x9) by A19,FUNCT_1:13 .= e by A11,A12,A15,A16,A18,Lm26; hence y in B; end; hence (f .: B) c= B by TARSKI:def 3; end; end; then A20: B is_stable_under_the_action_of A by Def1; now set F=<*>O; let x; assume A21: x in X; then reconsider e=x as Element of E; reconsider x9=e as Element of X by A21; len F = 0; then Product(F,A).x = (id E).x by Def3 .= x by A21,FUNCT_1:18; then Product(F,A).x9 = e; hence x in B; end; then X c= B by TARSKI:def 3; then A22: B = the_stable_subset_generated_by(X,A) by A20,A2,Def2; hereby assume a in the_stable_subset_generated_by(X,A); then consider e be Element of E such that A23: a=e and A24: ex F being FinSequence of O, x being Element of X st Product(F,A) .x = e by A22; consider F be FinSequence of O, x be Element of X such that A25: Product(F,A).x = e by A24; take F, x; thus Product(F,A).x = a by A23,A25; end; given F be FinSequence of O, x be Element of X such that A26: Product(F,A).x = a; thus thesis by A22,A26; end; :: GROUP_4:40 theorem Th25: for H being strict StableSubgroup of G holds the_stable_subgroup_of carr H = H proof let H be strict StableSubgroup of G; for H1 be strict StableSubgroup of G st carr H c= the carrier of H1 holds H is StableSubgroup of H1 by Lm21; hence thesis by Def26; end; :: GROUP_4:41 theorem Th26: A c= B implies the_stable_subgroup_of A is StableSubgroup of the_stable_subgroup_of B proof assume A1: A c= B; per cases; suppose A2: A is empty; reconsider H1 = (1).G,H2 = (1).(the_stable_subgroup_of B) as strict StableSubgroup of G by Th11; the carrier of H1 = {1_G} by Def8 .= {1_(the_stable_subgroup_of B)} by Th4 .= the carrier of H2 by Def8; then (1).G = (1).(the_stable_subgroup_of B) by Lm5; hence thesis by A2,Lm25; end; suppose A3: A is not empty; now set D = the_stable_subset_generated_by (B, the action of G); let a be Element of G; assume a in the_stable_subgroup_of A; then consider F be FinSequence of the carrier of G, I be FinSequence of INT, C be Subset of G such that A4: C = the_stable_subset_generated_by (A, the action of G) and A5: len F = len I and A6: rng F c= C and A7: Product(F |^ I) = a by Th24; now let y be set; assume A8: y in C; then reconsider b=y as Element of G; consider F1 be FinSequence of O, x be Element of A such that A9: Product(F1,the action of G).x = b by A3,A4,A8,Lm31; x in A by A3; hence y in D by A1,A9,Lm31; end; then C c= D by TARSKI:def 3; then rng F c= D by A6,XBOOLE_1:1; hence a in the_stable_subgroup_of B by A5,A7,Th24; end; hence thesis by Th13; end; end; scheme MeetSbgWOpEx{O() -> set, G() -> GroupWithOperators of O(), P[set]}: ex H being strict StableSubgroup of G() st the carrier of H = meet{A where A is Subset of G() : ex K being strict StableSubgroup of G() st A = the carrier of K & P[K]} provided A1: ex H being strict StableSubgroup of G() st P[H] proof set X = {A where A is Subset of G(): ex K being strict StableSubgroup of G() st A = the carrier of K & P[K]}; consider T being strict StableSubgroup of G() such that A2: P[T] by A1; A3: carr T in X by A2; then reconsider Y = meet X as Subset of G() by SETFAM_1:7; A4: now let a be Element of G(); assume A5: a in Y; now let Z be set; assume A6: Z in X; then consider A being Subset of G() such that A7: A = Z and A8: ex H being strict StableSubgroup of G() st A = the carrier of H & P[H]; consider H being StableSubgroup of G() such that A9: A = the carrier of H and P[H] by A8; a in the carrier of H by A5,A6,A7,A9,SETFAM_1:def 1; then a in H by STRUCT_0:def 5; then a" in H by Lm20; hence a" in Z by A7,A9,STRUCT_0:def 5; end; hence a" in Y by A3,SETFAM_1:def 1; end; A10: now let a,b be Element of G(); assume that A11: a in Y and A12: b in Y; now let Z be set; assume A13: Z in X; then consider A being Subset of G() such that A14: A = Z and A15: ex H being strict StableSubgroup of G() st A = the carrier of H & P[H]; consider H being StableSubgroup of G() such that A16: A = the carrier of H and P[H] by A15; b in the carrier of H by A12,A13,A14,A16,SETFAM_1:def 1; then A17: b in H by STRUCT_0:def 5; a in the carrier of H by A11,A13,A14,A16,SETFAM_1:def 1; then a in H by STRUCT_0:def 5; then a * b in H by A17,Lm19; hence a * b in Z by A14,A16,STRUCT_0:def 5; end; hence a * b in Y by A3,SETFAM_1:def 1; end; A18: now let o be Element of O(); let a be Element of G(); assume A19: a in Y; now let Z be set; assume A20: Z in X; then consider A being Subset of G() such that A21: A = Z and A22: ex H being strict StableSubgroup of G() st A = the carrier of H & P[H]; consider H being StableSubgroup of G() such that A23: A = the carrier of H and P[H] by A22; a in the carrier of H by A19,A20,A21,A23,SETFAM_1:def 1; then a in H by STRUCT_0:def 5; then (G()^o).a in H by Lm10; hence (G()^o).a in Z by A21,A23,STRUCT_0:def 5; end; hence (G()^o).a in Y by A3,SETFAM_1:def 1; end; now let Z be set; assume Z in X; then consider A being Subset of G() such that A24: Z = A and A25: ex K being strict StableSubgroup of G() st A = the carrier of K & P[K]; consider H being StableSubgroup of G() such that A26: A = the carrier of H and P[H] by A25; 1_G() in H by Lm18; hence 1_G() in Z by A24,A26,STRUCT_0:def 5; end; then Y <> {} by A3,SETFAM_1:def 1; hence thesis by A10,A4,A18,Lm15; end; :: GROUP_4:43 theorem Th27: the carrier of the_stable_subgroup_of A = meet{B where B is Subset of G: ex H being strict StableSubgroup of G st B = the carrier of H & A c= carr H} proof defpred P[StableSubgroup of G] means A c= carr $1; set X = {B where B is Subset of G :ex H being strict StableSubgroup of G st B = the carrier of H & A c= carr H}; A1: now let Y be set; assume Y in X; then ex B being Subset of G st Y = B & ex H being strict StableSubgroup of G st B = the carrier of H & A c= carr H; hence A c= Y; end; the carrier of (Omega).G = carr (Omega).G; then A2: ex H being strict StableSubgroup of G st P[H]; consider H being strict StableSubgroup of G such that A3: the carrier of H = meet{B where B is Subset of G: ex H being strict StableSubgroup of G st B = the carrier of H & P[H]} from MeetSbgWOpEx(A2); A4: now let H1 be strict StableSubgroup of G; A5: the carrier of H1 = carr H1; assume A c= the carrier of H1; then the carrier of H1 in X by A5; hence H is StableSubgroup of H1 by A3,Lm21,SETFAM_1:3; end; carr (Omega).G in X; then A c= the carrier of H by A3,A1,SETFAM_1:5; hence thesis by A3,A4,Def26; end; Lm32: B = the carrier of gr A implies the_stable_subgroup_of A = the_stable_subgroup_of B proof A1: A c= the carrier of gr A by GROUP_4:def 4; assume A2: B = the carrier of gr A; A3: now let H be strict StableSubgroup of G; reconsider H9=the multMagma of H as strict Subgroup of G by Lm16; assume A c= the carrier of H; then gr A is Subgroup of H9 by GROUP_4:def 4; then B c= the carrier of H9 by A2,GROUP_2:def 5; hence the_stable_subgroup_of B is StableSubgroup of H by Def26; end; the carrier of gr A c= the carrier of the_stable_subgroup_of B by A2,Def26; then A c= the carrier of the_stable_subgroup_of B by A1,XBOOLE_1:1; hence thesis by A3,Def26; end; :: GROUP_4:64 theorem Th28: for N1,N2 being strict normal StableSubgroup of G holds N1 * N2 = N2 * N1 proof let N1,N2 be strict normal StableSubgroup of G; reconsider N19= the multMagma of N1,N29= the multMagma of N2 as strict normal Subgroup of G by Lm7; thus N1 * N2 = carr N29 * carr N19 by GROUP_3:125 .= N2 * N1; end; :: GROUP_4:68 theorem Th29: H1 "\/" H2 = the_stable_subgroup_of(H1 * H2) proof reconsider H19=H1,H29=H2 as Subgroup of G by Def7; reconsider Y = the carrier of (H19"\/"H29) as Subset of G by GROUP_2:def 5; A1: Y = the carrier of gr(H19*H29) by GROUP_4:50; H1 "\/" H2 = the_stable_subgroup_of Y by Lm32 .= the_stable_subgroup_of(H19*H29) by A1,Lm32; hence thesis; end; :: GROUP_4:69 theorem Th30: H1 * H2 = H2 * H1 implies the carrier of H1 "\/" H2 = H1 * H2 proof assume H1 * H2 = H2 * H1; then consider H being strict StableSubgroup of G such that A1: the carrier of H = carr H1 * carr H2 by Th17; now set A = carr H1 \/ carr H2; let a be Element of G; set X = {B where B is Subset of G: ex H being strict StableSubgroup of G st B = the carrier of H & A c= carr H}; assume a in H; then a in carr H1 * carr H2 by A1,STRUCT_0:def 5; then consider b,c be Element of G such that A2: a = b * c and A3: b in carr H1 and A4: c in carr H2; A5: now let Y be set; assume Y in X; then consider B be Subset of G such that A6: Y = B and A7: ex H being strict StableSubgroup of G st B = the carrier of H & A c= carr H; consider H9 be strict StableSubgroup of G such that A8: B = the carrier of H9 and A9: A c= carr H9 by A7; c in A by A4,XBOOLE_0:def 3; then A10: c in H9 by A9,STRUCT_0:def 5; A11: H9 is Subgroup of G by Def7; b in A by A3,XBOOLE_0:def 3; then b in H9 by A9,STRUCT_0:def 5; then b * c in H9 by A11,A10,GROUP_2:50; hence a in Y by A2,A6,A8,STRUCT_0:def 5; end; carr (Omega).G in X; then a in meet X by A5,SETFAM_1:def 1; then a in the carrier of the_stable_subgroup_of A by Th27; hence a in H1 "\/" H2 by STRUCT_0:def 5; end; then H is StableSubgroup of H1 "\/" H2 by Th13; then H is Subgroup of H1 "\/" H2 by Def7; then A12: the carrier of H c= the carrier of H1 "\/" H2 by GROUP_2:def 5; carr H1 \/ carr H2 c= carr H1 * carr H2 proof let x; assume A13: x in carr H1 \/ carr H2; then reconsider a = x as Element of G; now per cases by A13,XBOOLE_0:def 3; suppose A14: x in carr H1; 1_G in H2 by Lm18; then A15: 1_G in carr H2 by STRUCT_0:def 5; a * 1_G = a by GROUP_1:def 4; hence thesis by A14,A15; end; suppose A16: x in carr H2; 1_G in H1 by Lm18; then A17: 1_G in carr H1 by STRUCT_0:def 5; 1_G * a = a by GROUP_1:def 4; hence thesis by A16,A17; end; end; hence thesis; end; then H1 "\/" H2 is StableSubgroup of H by A1,Def26; then H1 "\/" H2 is Subgroup of H by Def7; then the carrier of H1 "\/" H2 c= the carrier of H by GROUP_2:def 5; hence thesis by A1,A12,XBOOLE_0:def 10; end; :: GROUP_4:71 theorem Th31: for N1,N2 being strict normal StableSubgroup of G holds the carrier of N1 "\/" N2 = N1 * N2 proof let N1,N2 be strict normal StableSubgroup of G; N1 * N2 = N2 * N1 by Th28; hence thesis by Th30; end; :: GROUP_4:72 theorem Th32: for N1,N2 being strict normal StableSubgroup of G holds N1 "\/" N2 is normal StableSubgroup of G proof let N1,N2 be strict normal StableSubgroup of G; (ex N be strict normal StableSubgroup of G st the carrier of N = carr N1 * carr N2 )& the carrier of N1 "\/" N2 = N1 * N2 by Th23,Th31; hence thesis by Lm5; end; :: GROUP_4:76 theorem Th33: for H being strict StableSubgroup of G holds (1).G "\/" H = H & H "\/" (1).G = H proof let H be strict StableSubgroup of G; 1_G in H by Lm18; then 1_G in carr H by STRUCT_0:def 5; then {1_G} c= carr H by ZFMISC_1:31; then A1: {1_G} \/ carr H = carr H by XBOOLE_1:12; carr(1).G = {1_G} by Def8; hence thesis by A1,Th25; end; :: GROUP_4:77 theorem Th34: (Omega).G "\/" H1 = (Omega).G & H1 "\/" (Omega).G = (Omega).G proof (the carrier of (Omega).G) \/ carr H1 = [#] the carrier of G by SUBSET_1:11; hence thesis by Th25; end; :: Lm7 in GROUP_4 Lm33: H1 is StableSubgroup of H1 "\/" H2 proof carr H1 c= carr H1 \/ carr H2 & carr H1 \/ carr H2 c= the carrier of the_stable_subgroup_of (carr H1 \/ carr H2) by Def26,XBOOLE_1:7; hence thesis by Lm21,XBOOLE_1:1; end; :: GROUP_4:78 theorem Th35: H1 is StableSubgroup of H1 "\/" H2 & H2 is StableSubgroup of H1 "\/" H2 proof H1 "\/" H2 = H2 "\/" H1; hence thesis by Lm33; end; :: GROUP_4:79 theorem Th36: for H2 being strict StableSubgroup of G holds H1 is StableSubgroup of H2 iff H1 "\/" H2 = H2 proof let H2 be strict StableSubgroup of G; thus H1 is StableSubgroup of H2 implies H1 "\/" H2 = H2 proof assume H1 is StableSubgroup of H2; then H1 is Subgroup of H2 by Def7; then the carrier of H1 c= the carrier of H2 by GROUP_2:def 5; hence H1 "\/" H2 = the_stable_subgroup_of carr H2 by XBOOLE_1:12 .= H2 by Th25; end; thus thesis by Th35; end; :: GROUP_4:81 theorem Th37: for H3 being strict StableSubgroup of G holds H1 is StableSubgroup of H3 & H2 is StableSubgroup of H3 implies H1 "\/" H2 is StableSubgroup of H3 proof let H3 be strict StableSubgroup of G; assume that A1: H1 is StableSubgroup of H3 and A2: H2 is StableSubgroup of H3; H2 is Subgroup of H3 by A2,Def7; then A3: carr H2 c= carr H3 by GROUP_2:def 5; H1 is Subgroup of H3 by A1,Def7; then carr H1 c= carr H3 by GROUP_2:def 5; then the_stable_subgroup_of(carr H1 \/ carr H2) is StableSubgroup of the_stable_subgroup_of carr H3 by A3,Th26,XBOOLE_1:8; hence thesis by Th25; end; :: GROUP_4:82 theorem Th38: for H2,H3 being strict StableSubgroup of G holds H1 is StableSubgroup of H2 implies H1 "\/" H3 is StableSubgroup of H2 "\/" H3 proof let H2,H3 be strict StableSubgroup of G; assume H1 is StableSubgroup of H2; then H1 is Subgroup of H2 by Def7; then carr H1 c= carr H2 by GROUP_2:def 5; hence thesis by Th26,XBOOLE_1:9; end; :: GROUP_6:3 theorem Th39: for X,Y being StableSubgroup of H1, X9,Y9 being StableSubgroup of G st X = X9 & Y = Y9 holds X9 /\ Y9 = X /\ Y proof let X,Y be StableSubgroup of H1; reconsider Z = X /\ Y as StableSubgroup of G by Th11; let X9,Y9 be StableSubgroup of G; assume A1: X=X9 & Y=Y9; the carrier of X /\ Y = (carr X) /\ (carr Y) by Def25; then X9 /\ Y9 = Z by A1,Th18; hence thesis; end; :: GROUP_6:9 theorem Th40: N is StableSubgroup of H1 implies N is normal StableSubgroup of H1 proof assume N is StableSubgroup of H1; then reconsider N9 = N as StableSubgroup of H1; now reconsider N99=the multMagma of N as normal Subgroup of G by Lm7; let H be strict Subgroup of H1; assume A1: H = the multMagma of N9; reconsider N as Subgroup of G by Def7; H1 is Subgroup of G & N99 is Subgroup of N by Def7,GROUP_2:57; hence H is normal by A1,GROUP_6:8; end; hence thesis by Def10; end; :: Lm4 in GROUP_6 Lm34: H1 /\ H2 is StableSubgroup of H1 proof the carrier of H1 /\ H2 = (the carrier of H1) /\ (the carrier of H2) by Th18; hence thesis by Lm21,XBOOLE_1:17; end; :: GROUP_6:10 theorem Th41: H1 /\ N is normal StableSubgroup of H1 & N /\ H1 is normal StableSubgroup of H1 proof thus H1 /\ N is normal StableSubgroup of H1 proof reconsider A = H1 /\ N as StableSubgroup of H1 by Lm34; now reconsider N9=the multMagma of N as normal Subgroup of G by Lm7; let H be strict Subgroup of H1; assume A1: H = the multMagma of A; now let b be Element of H1; thus b * H c= H * b proof let x; assume x in b * H; then consider a be Element of H1 such that A2: x = b * a and A3: a in H by GROUP_2:103; reconsider a9 = a, b9 = b as Element of G by Th2; reconsider x9 = x as Element of H1 by A2; A4: b9" = b" by Th6; a in the carrier of A by A1,A3,STRUCT_0:def 5; then a in carr(H1) /\ carr(N) by Def25; then a in carr N9 by XBOOLE_0:def 4; then A5: a in N9 by STRUCT_0:def 5; x = b9 * a9 by A2,Th3; then A6: x in b9 * N9 by A5,GROUP_2:103; b9 * N9 c= N9 * b9 by GROUP_3:118; then consider b1 be Element of G such that A7: x = b1 * b9 and A8: b1 in N9 by A6,GROUP_2:104; reconsider x99 = x as Element of G by A7; b1 = x99 * b9" by A7,GROUP_1:14; then A9: b1 = x9 * b" by A4,Th3; then reconsider b19 = b1 as Element of H1; b1 in the carrier of N by A8,STRUCT_0:def 5; then b1 in carr(H1) /\ carr(N) by A9,XBOOLE_0:def 4; then b19 in the carrier of A by Def25; then A10: b19 in H by A1,STRUCT_0:def 5; b19 * b = x by A7,Th3; hence thesis by A10,GROUP_2:104; end; end; hence H is normal by GROUP_3:118; end; hence thesis by Def10; end; hence thesis; end; :: GROUP_6:13 theorem Th42: for G being strict GroupWithOperators of O holds G is trivial implies (1).G = G proof let G be strict GroupWithOperators of O; reconsider H=G as StableSubgroup of G by Lm4; assume G is trivial; then ex x be set st the carrier of G = {x} by ZFMISC_1:131; then the carrier of H = {1_G} by TARSKI:def 1; hence thesis by Def8; end; Lm35: for N9 being normal Subgroup of G st N9 = the multMagma of N holds G./. N9 = the multMagma of G./.N & 1_(G./.N9) = 1_(G./.N) proof let N9 be normal Subgroup of G; assume A1: N9 = the multMagma of N; then reconsider e=1_(G./.N9) as Element of G./.N by Def14; Cosets N9 = Cosets N by A1,Def14; hence G./.N9 = the multMagma of G./.N by A1,Def15; now let h be Element of G./.N; reconsider h9=h as Element of G./.N9 by A1,Def14; thus h * e = h9 * 1_(G./.N9) by A1,Def15 .= h by GROUP_1:def 4; thus e * h = 1_(G./.N9) * h9 by A1,Def15 .= h by GROUP_1:def 4; end; hence thesis by GROUP_1:4; end; :: GROUP_6:29 theorem Th43: 1_(G./.N) = carr N proof reconsider N9 = the multMagma of N as normal Subgroup of G by Lm7; 1_(G./.N9) = carr N9 by GROUP_6:24; hence thesis by Lm35; end; :: GROUP_6:35 theorem Th44: for M,N being strict normal StableSubgroup of G, MN being normal StableSubgroup of N st MN=M & M is StableSubgroup of N holds N./.MN is normal StableSubgroup of G./.M proof let M,N be strict normal StableSubgroup of G; reconsider M9 = the multMagma of M as normal Subgroup of G by Lm7; reconsider N9 = the multMagma of N as normal Subgroup of G by Lm7; let MN be normal StableSubgroup of N; assume A1: MN=M; reconsider MN99=(N9,M9)`*` as normal Subgroup of N9; reconsider MN9 = the multMagma of MN as normal Subgroup of N by Lm7; assume M is StableSubgroup of N; then M is Subgroup of N by Def7; then the carrier of M c= the carrier of N & the multF of M = (the multF of N) || the carrier of M by GROUP_2:def 5; then A2: M9 is Subgroup of N9 by GROUP_2:def 5; then A3: (N9,M9)`*` = MN9 by A1,GROUP_6:def 1; reconsider K=N9./.(N9,M9)`*` as normal Subgroup of G./.M9 by A2,GROUP_6:29; A4: now let x; hereby assume x in Cosets MN9; then consider a be Element of N such that A5: x = a * MN9 and x = MN9 * a by GROUP_6:13; reconsider a9 = a as Element of N9; reconsider A = {a} as Subset of N by ZFMISC_1:31; reconsider A9 = {a9} as Subset of N9 by ZFMISC_1:31; now let y be set; hereby assume y in {g * h where g,h is Element of N:g in A & h in carr MN9}; then consider g,h be Element of N such that A6: y = g*h and A7: g in A & h in carr MN9; reconsider h9=h as Element of N9; reconsider g9=g as Element of N9; y = g9*h9 by A6; hence y in {g99*h99 where g99,h99 is Element of N9: g99 in A9 & h99 in carr MN99} by A3,A7; end; assume y in {g*h where g,h is Element of N9: g in A9 & h in carr MN99}; then consider g,h be Element of N9 such that A8: y = g*h and A9: g in A9 & h in carr MN99; reconsider h9=h as Element of N; reconsider g9=g as Element of N; y = g9*h9 by A8; hence y in {g99*h99 where g99,h99 is Element of N: g99 in A & h99 in carr MN9} by A3,A9; end; then x = a9 * MN99 by A5,TARSKI:1; hence x in Cosets MN99 by GROUP_6:14; end; assume x in Cosets MN99; then consider a9 be Element of N9 such that A10: x = a9 * MN99 and x = MN99 * a9 by GROUP_6:13; reconsider a = a9 as Element of N; reconsider A9 = {a9} as Subset of N9 by ZFMISC_1:31; reconsider A = {a} as Subset of N by ZFMISC_1:31; now let y be set; hereby assume y in {g * h where g,h is Element of N:g in A & h in carr MN9}; then consider g,h be Element of N such that A11: y = g*h and A12: g in A & h in carr MN9; reconsider h9=h as Element of N9; reconsider g9=g as Element of N9; y = g9*h9 by A11; hence y in {g99*h99 where g99,h99 is Element of N9: g99 in A9 & h99 in carr MN99} by A3,A12; end; assume y in {g*h where g,h is Element of N9: g in A9 & h in carr MN99}; then consider g,h be Element of N9 such that A13: y = g*h and A14: g in A9 & h in carr MN99; reconsider h9=h as Element of N; reconsider g9=g as Element of N; y = g9*h9 by A13; hence y in {g99*h99 where g99,h99 is Element of N: g99 in A & h99 in carr MN9} by A3,A14; end; then x = a * MN9 by A10,TARSKI:1; hence x in Cosets MN9 by GROUP_6:14; end; then A15: the carrier of K = Cosets MN9 by TARSKI:1 .= the carrier of N./.MN by Def14; A16: now let H be strict Subgroup of G./.M; assume A17: H = the multMagma of N./.MN; now let a be Element of G./.M; reconsider a9=a as Element of G./.M9 by Def14; now let x; assume x in a * carr H; then consider b be Element of G./.M such that A18: x = a * b and A19: b in carr H by GROUP_2:27; reconsider b9=b as Element of G./.M9 by Def14; A20: x = a9 * b9 by A18,Def15; then reconsider x9=x as Element of G./.M9; a9 * K c= K * a9 & x9 in a9 * carr K by A15,A17,A19,A20,GROUP_2:27 ,GROUP_3:118; then consider c9 be Element of G./.M9 such that A21: x9 = c9 * a9 and A22: c9 in carr K by GROUP_2:28; reconsider c = c9 as Element of G./.M by Def14; x = c * a by A21,Def15; hence x in carr H * a by A15,A17,A22,GROUP_2:28; end; hence a * H c= H * a by TARSKI:def 3; end; hence H is normal by GROUP_3:118; end; A23: the carrier of G./.M = the carrier of G./.M9 by Def14; then A24: the carrier of N./.MN c= the carrier of G./.M by A15,GROUP_2:def 5; A25: now let o be Element of O; per cases; suppose A26: not o in O; A27: the carrier of N./.MN c= the carrier of G./.M by A23,A15,GROUP_2:def 5; A28: now let x,y be set; assume A29: [x,y] in id the carrier of N./.MN; then A30: x in the carrier of N./.MN by RELAT_1:def 10; x=y by A29,RELAT_1:def 10; then [x,y] in id the carrier of G./.M by A27,A30,RELAT_1:def 10; hence [x,y] in (id the carrier of G./.M)|the carrier of N./.MN by A30, RELAT_1:def 11; end; A31: now let x,y be set; assume A32: [x,y] in (id the carrier of G./.M)|the carrier of N./.MN; then [x,y] in id the carrier of G ./.M by RELAT_1:def 11; then A33: x=y by RELAT_1:def 10; x in the carrier of N./.MN by A32,RELAT_1:def 11; hence [x,y] in id the carrier of N./.MN by A33,RELAT_1:def 10; end; thus (N./.MN)^o = id the carrier of N./.MN by A26,Def6 .= (id the carrier of G./.M)|the carrier of N./.MN by A28,A31, RELAT_1:def 2 .= ((G./.M)^o)|the carrier of N./.MN by A26,Def6; end; suppose A34: o in O; then (the action of G./.M).o in Funcs(the carrier of G./.M, the carrier of G./.M) by FUNCT_2:5; then consider f be Function such that A35: f=(the action of G./.M).o and A36: dom f = the carrier of G./.M and rng f c= the carrier of G./.M by FUNCT_2:def 2; A37: f = {[A,B] where A,B is Element of Cosets M: ex a,b being Element of G st a in A & b in B & b = (G^o).a} by A34,A35,Def16; (the action of N./.MN).o in Funcs(the carrier of N./.MN, the carrier of N./.MN) by A34,FUNCT_2:5; then consider g be Function such that A38: g=(the action of N./.MN).o and A39: dom g = the carrier of N./.MN and rng g c= the carrier of N./.MN by FUNCT_2:def 2; A40: dom g = dom f /\ the carrier of N./.MN by A24,A36,A39,XBOOLE_1:28; A41: g = {[A,B] where A,B is Element of Cosets MN: ex a,b being Element of N st a in A & b in B & b = (N^o).a} by A34,A38,Def16; A42: now let x; assume A43: x in dom g; then [x,g.x] in g by FUNCT_1:1; then consider A2,B2 be Element of Cosets MN such that A44: [x,g.x]=[A2,B2] and A45: ex a,b being Element of N st a in A2 & b in B2 & b = (N^o).a by A41; A46: A2=x by A44,XTUPLE_0:1; [x,f.x] in f by A24,A36,A39,A43,FUNCT_1:1; then consider A1,B1 be Element of Cosets M such that A47: [x,f.x]=[A1,B1] and A48: ex a,b being Element of G st a in A1 & b in B1 & b = (G^o).a by A37; A49: A1=x by A47,XTUPLE_0:1; reconsider A29=A2,B29=B2 as Element of Cosets MN9 by Def14; reconsider A19=A1,B19=B1 as Element of Cosets M9 by Def14; set fo = G^o; N is Subgroup of G by Def7; then A50: the carrier of N c= the carrier of G by GROUP_2:def 5; consider a2,b2 be Element of N such that A51: a2 in A2 and A52: b2 in B2 and A53: b2 = (N^o).a2 by A45; A54: B29 = b2 * MN9 by A52,Lm9; a2 in the carrier of N & b2 in the carrier of N; then reconsider a29=a2,b29=b2 as Element of G by A50; consider a1,b1 be Element of G such that A55: a1 in A1 and A56: b1 in B1 and A57: b1 = (G^o).a1 by A48; A58: A19 = a1 * M9 by A55,Lm9; now let x; hereby assume x in b2 * carr MN9; then consider h be Element of N such that A59: x = b2 * h and A60: h in carr MN9 by GROUP_2:27; h in the carrier of N; then reconsider h9=h as Element of G by A50; x = b29 * h9 by A59,Th3; hence x in b29 * carr M9 by A1,A60,GROUP_2:27; end; assume x in b29 * carr M9; then consider h be Element of G such that A61: x = b29 * h and A62: h in carr M9 by GROUP_2:27; h in carr MN9 by A1,A62; then reconsider h9=h as Element of N; x = b2 * h9 by A61,Th3; hence x in b2 * carr MN9 by A1,A62,GROUP_2:27; end; then A63: b29 * M9 = b2 * MN9 by TARSKI:1; A64: B2=g.x by A44,XTUPLE_0:1; A65: B1=f.x by A47,XTUPLE_0:1; now let x; hereby assume x in a2 * carr MN9; then consider h be Element of N such that A66: x = a2 * h and A67: h in carr MN9 by GROUP_2:27; h in the carrier of N; then reconsider h9=h as Element of G by A50; x = a29 * h9 by A66,Th3; hence x in a29 * carr M9 by A1,A67,GROUP_2:27; end; assume x in a29 * carr M9; then consider h be Element of G such that A68: x = a29 * h and A69: h in carr M9 by GROUP_2:27; h in carr MN9 by A1,A69; then reconsider h9=h as Element of N; x = a2 * h9 by A68,Th3; hence x in a2 * carr MN9 by A1,A69,GROUP_2:27; end; then A70: a2 * MN9 = a29 * M9 by TARSKI:1; A29 = a2 * MN9 by A51,Lm9; then a1" * a29 in M9 by A49,A46,A58,A70,GROUP_2:114; then a1" * a29 in the carrier of M by STRUCT_0:def 5; then a1" * a29 in M by STRUCT_0:def 5; then A71: fo.(a1" * a29) in M by Lm10; A72: b1" = fo.a1" by A57,GROUP_6:32; b29 = ((G^o)|the carrier of N).a2 by A53,Def7 .= fo.a29 by FUNCT_1:49; then b1" * b29 in M by A72,A71,GROUP_6:def 6; then b1" * b29 in the carrier of M by STRUCT_0:def 5; then A73: b1" * b29 in M9 by STRUCT_0:def 5; B19 = b1 * M9 by A56,Lm9; hence g.x = f.x by A65,A64,A63,A73,A54,GROUP_2:114; end; thus (N./.MN)^o = (the action of N./.MN).o by A34,Def6 .= f|the carrier of N./.MN by A38,A40,A42,FUNCT_1:46 .= ((G./.M)^o)|the carrier of N./.MN by A34,A35,Def6; end; end; Cosets MN99 = Cosets MN9 by A4,TARSKI:1; then reconsider f=CosOp MN99 as BinOp of Cosets MN9; now let W1,W2 be Element of Cosets MN9; reconsider W19=W1,W29=W2 as Element of Cosets MN99 by A4; let A1,A2 be Subset of N; assume A74: W1 = A1; reconsider A19=A1,A29=A2 as Subset of N9; assume A75: W2 = A2; A76: now let x; hereby assume x in A1 * A2; then consider g,h be Element of N such that A77: x = g * h and A78: g in A1 & h in A2; reconsider g9=g,h9=h as Element of N9; x = g9 * h9 by A77; hence x in A19 * A29 by A78; end; assume x in A19 * A29; then consider g9,h9 be Element of N9 such that A79: x = g9 * h9 and A80: g9 in A19 & h9 in A29; reconsider g=g9,h=h9 as Element of N; x = g * h by A79; hence x in A1 * A2 by A80; end; thus f.(W1,W2) = f.(W19,W29) .= A19 * A29 by A74,A75,GROUP_6:def 3 .= A1 * A2 by A76,TARSKI:1; end; then the multF of K = CosOp MN9 by GROUP_6:def 3 .= the multF of N./.MN by Def15; then the multF of N./.MN = (the multF of G./.M9)||the carrier of K by GROUP_2:def 5 .= (the multF of G./.M)||the carrier of N./.MN by A15,Def15; then N./.MN is Subgroup of G./.M by A24,GROUP_2:def 5; hence thesis by A16,A25,Def7,Def10; end; :: GROUP_6:40 theorem h.(1_G)=1_H by Lm13; :: GROUP_6:41 theorem h.(g1")=(h.g1)" by Lm14; :: GROUP_6:50 theorem Th47: g1 in Ker h iff h.g1 = 1_H proof thus g1 in Ker h implies h.g1 = 1_H proof assume g1 in Ker h; then g1 in the carrier of Ker h by STRUCT_0:def 5; then g1 in {b where b is Element of G : h.b = 1_H} by Def21; then ex b being Element of G st g1 = b & h.b = 1_H; hence thesis; end; assume h.g1 = 1_H; then g1 in {b where b is Element of G: h.b = 1_H}; then g1 in the carrier of Ker h by Def21; hence thesis by STRUCT_0:def 5; end; :: GROUP_6:52 theorem Th48: for N being strict normal StableSubgroup of G holds Ker nat_hom N = N proof let N be strict normal StableSubgroup of G; reconsider N9 = the multMagma of N as strict normal Subgroup of G by Lm7; A1: nat_hom N = nat_hom N9 & 1_(G./.N) = 1_(G./.N9) by Def20,Lm35; the carrier of Ker nat_hom N = {a where a is Element of G: (nat_hom N).a = 1_(G./.N)} by Def21 .= {a where a is Element of G: (nat_hom N9).a = 1_(G./.N9)} by A1 .= the carrier of Ker nat_hom N9 by GROUP_6:def 9 .= the carrier of N by GROUP_6:43; hence thesis by Lm5; end; :: GROUP_6:53 theorem Th49: rng h = the carrier of Image h proof the carrier of Image h = h .: (the carrier of G) by Def22 .= h .: (dom h) by FUNCT_2:def 1 .= rng h by RELAT_1:113; hence thesis; end; :: GROUP_6:57 theorem Th50: Image nat_hom N = G./.N proof reconsider N9 = the multMagma of N as strict normal Subgroup of G by Lm7; reconsider H = G./.N as strict StableSubgroup of G./.N by Lm4; A1: G./.N9 = the multMagma of G./.N by Lm35; the carrier of Image nat_hom N = nat_hom N .: (the carrier of G) by Def22 .= nat_hom N9 .: (the carrier of G) by Def20 .= the carrier of Image nat_hom N9 by GROUP_6:def 10 .= the carrier of H by A1,GROUP_6:48; hence thesis by Lm5; end; :: GROUP_6:67 theorem Th51: for H being strict GroupWithOperators of O, h being Homomorphism of G,H holds h is onto iff Image h = H proof let H be strict GroupWithOperators of O, h be Homomorphism of G,H; thus h is onto implies Image h = H proof reconsider H9=H as strict StableSubgroup of H by Lm4; assume rng h = the carrier of H; then the carrier of H9 = the carrier of Image h by Th49; hence thesis by Lm5; end; assume A1: Image h = H; the carrier of H c= rng h proof let x; assume x in the carrier of H; then x in h .: (the carrier of G) by A1,Def22; then ex y being set st y in dom h & y in the carrier of G & h.y = x by FUNCT_1:def 6; hence thesis by FUNCT_1:def 3; end; then rng h = the carrier of H by XBOOLE_0:def 10; hence thesis by FUNCT_2:def 3; end; :: GROUP_6:68 theorem Th52: for H being strict GroupWithOperators of O, h being Homomorphism of G,H st h is onto holds for c being Element of H ex a being Element of G st h .a = c proof let H be strict GroupWithOperators of O; let h be Homomorphism of G,H; assume A1: h is onto; let c be Element of H; rng h = the carrier of H by A1,FUNCT_2:def 3; then consider a be set such that A2: a in dom h and A3: c = h.a by FUNCT_1:def 3; reconsider a as Element of G by A2; take a; thus thesis by A3; end; :: GROUP_6:69 theorem Th53: nat_hom N is onto proof Image nat_hom N = G./.N by Th50; hence thesis by Th51; end; :: GROUP_6:75 theorem Th54: nat_hom (1).G is bijective proof reconsider H = the multMagma of (1).G as strict normal Subgroup of G by Lm7; set g = nat_hom (1).G; reconsider G9=G as Group; A1: the carrier of H = {1_G9} by Def8; A2: nat_hom (1).G9 is bijective & g is onto by Th53,GROUP_6:65; nat_hom (1).G = nat_hom H by Def20 .= nat_hom (1).G9 by A1,GROUP_2:def 7; hence thesis by A2; end; :: GROUP_6:78 theorem Th55: G,H are_isomorphic & H,I are_isomorphic implies G,I are_isomorphic proof assume that A1: G,H are_isomorphic and A2: H,I are_isomorphic; consider g be Homomorphism of G,H such that A3: g is bijective by A1,Def19; consider h1 be Homomorphism of H,I such that A4: h1 is bijective by A2,Def19; A5: rng h1 = the carrier of I by A4,FUNCT_2:def 3; rng g = the carrier of H by A3,FUNCT_2:def 3; then dom h1 = rng g by FUNCT_2:def 1; then rng(h1 * g) = the carrier of I by A5,RELAT_1:28; then h1 * g is onto by FUNCT_2:def 3; hence thesis by A3,A4,Def19; end; :: GROUP_6:82 theorem Th56: for G being strict GroupWithOperators of O holds G,G./.(1).G are_isomorphic proof let G be strict GroupWithOperators of O; nat_hom (1).G is bijective by Th54; hence thesis by Def19; end; :: GROUP_6:83 theorem Th57: for G being strict GroupWithOperators of O holds G./.(Omega).G is trivial proof let G be strict GroupWithOperators of O; reconsider G9=G as Group; reconsider H=the multMagma of (Omega).G as strict normal Subgroup of G by Lm7 ; A1: H = (Omega).G9; the carrier of G./.(Omega).G = Cosets H by Def14 .= {the carrier of G} by A1,GROUP_2:142; hence thesis; end; :: GROUP_6:87 theorem Th58: for G,H being strict GroupWithOperators of O holds G,H are_isomorphic & G is trivial implies H is trivial proof let G,H be strict GroupWithOperators of O; assume that A1: G,H are_isomorphic and A2: G is trivial; consider e be set such that A3: the carrier of G = {e} by A2,GROUP_6:def 2; consider g be Homomorphism of G,H such that A4: g is bijective by A1,Def19; e in the carrier of G by A3,TARSKI:def 1; then A5: e in dom g by FUNCT_2:def 1; the carrier of H = the carrier of Image g by A4,Th51 .= Im(g,e) by A3,Def22 .= {g.e} by A5,FUNCT_1:59; hence thesis; end; :: GROUP_6:90 theorem Th59: G./.Ker h, Image h are_isomorphic proof reconsider G9=G,H9=H as Group; reconsider h9=h as Homomorphism of G9,H9; consider g9 be Homomorphism of G9./.Ker h9, Image h9 such that A1: g9 is bijective and A2: h9 = g9 * nat_hom Ker h9 by GROUP_6:79; A3: the carrier of Image h9 = h9 .: (the carrier of G9) by GROUP_6:def 10 .= the carrier of Image h by Def22; now let x; hereby assume x in the carrier of Ker h; then x in {a where a is Element of G: h.a = 1_H} by Def21; hence x in the carrier of Ker h9 by GROUP_6:def 9; end; assume x in the carrier of Ker h9; then x in {a9 where a9 is Element of G9: h9.a9 = 1_H9} by GROUP_6:def 9; hence x in the carrier of Ker h by Def21; end; then A4: the carrier of Ker h9 = the carrier of Ker h by TARSKI:1; Ker h is Subgroup of G by Def7; then A5: the multMagma of Ker h9 = the multMagma of Ker h by A4,GROUP_2:59; then the carrier of G9./.Ker h9 = the carrier of G./.Ker h by Def14; then reconsider g=g9 as Function of G./.Ker h, Image h by A3; Image h is Subgroup of H by Def7; then A6: the multMagma of Image h9 = the multMagma of Image h by A3,GROUP_2:59; A7: now let a, b be Element of G./.Ker h; reconsider b9=b as Element of G9./.Ker h9 by A5,Def14; reconsider a9=a as Element of G9./.Ker h9 by A5,Def14; thus g.(a * b) = g9.(a9 * b9) by A5,Def15 .= g9.a9 * g9.b9 by GROUP_6:def 6 .= g.a * g.b by A6; end; now let o be Element of O; let a be Element of G./.Ker h; per cases; suppose A8: O is empty; hence g.(((G./.Ker h)^o).a) = g.((id the carrier of (G./.Ker h)).a) by Def6 .= g.a by FUNCT_1:18 .= (id the carrier of Image h).(g.a) by FUNCT_1:18 .= ((Image h)^o).(g.a) by A8,Def6; end; suppose A9: O is not empty; reconsider G99=G./.Ker h as Group; set f = (the action of G./.Ker h).o; A10: f = {[A,B] where A,B is Element of Cosets Ker h: ex g,h being Element of G st g in A & h in B & h=(G^o).g} by A9,Def16; f = (G./.Ker h)^o by A9,Def6; then reconsider f as Homomorphism of G99, G99; a in the carrier of G99; then a in dom f by FUNCT_2:def 1; then [a,f.a] in f by FUNCT_1:1; then consider A,B be Element of Cosets Ker h such that A11: [A,B]=[a,f.a] and A12: ex g1,g2 being Element of G st g1 in A & g2 in B & g2=(G^o).g1 by A10; reconsider A,B as Element of Cosets Ker h9 by A5,Def14; consider g1,g2 be Element of G9 such that A13: g1 in A and A14: g2 in B and A15: g2=(G^o).g1 by A12; A16: A = g1 * Ker h9 by A13,Lm9; g1 in the carrier of G9; then A17: g1 in dom nat_hom Ker h9 by FUNCT_2:def 1; g2 in the carrier of G9; then A18: g2 in dom nat_hom Ker h9 by FUNCT_2:def 1; A19: ((Image h)^o).(g.a) = ((H^o)|the carrier of Image h).(g.a) by Def7 .= (H^o).(g.a) by FUNCT_1:49 .= (H^o).(g9.(g1 * Ker h9)) by A11,A16,XTUPLE_0:1; A20: B = g2 * Ker h9 by A14,Lm9; h9.g2 = (H^o).(h9.g1) by A15,Def18; then g9.(nat_hom Ker h9.g2)=(H^o).((g9 * nat_hom Ker h9).g1) by A2,A18, FUNCT_1:13; then g9.(nat_hom Ker h9.g2)=(H^o).(g9.(nat_hom Ker h9.g1)) by A17, FUNCT_1:13; then A21: g9.(g2 * Ker h9) = (H^o).(g9.(nat_hom Ker h9.g1)) by GROUP_6:def 8; g.(((G./.Ker h)^o).a) = g.(f.a) by A9,Def6 .= g9.(g2 * Ker h9) by A11,A20,XTUPLE_0:1; hence g.(((G./.Ker h)^o).a)=((Image h)^o).(g.a) by A19,A21,GROUP_6:def 8; end; end; then reconsider g as Homomorphism of G./.Ker h, Image h by A7,Def18, GROUP_6:def 6; g is onto by A1,A3; hence thesis by A1,Def19; end; :: GRSOLV_1:1 theorem Th60: for H,F1,F2 being strict StableSubgroup of G st F1 is normal StableSubgroup of F2 holds H /\ F1 is normal StableSubgroup of H /\ F2 proof let H,F1,F2 be strict StableSubgroup of G; reconsider F=F2 /\ H as StableSubgroup of F2 by Lm34; assume A1: F1 is normal StableSubgroup of F2; then A2: F1 /\ H=(F1 /\ F2) /\ H by Lm22 .=F1 /\ (F2 /\ H) by Th20; reconsider F1 as normal StableSubgroup of F2 by A1; F1 /\ F is normal StableSubgroup of F by Th41; hence thesis by A2,Th39; end; begin :: Other Theorems on Actions and Groups with Operators reserve E for set, A for Action of O,E, C for Subset of G, N1 for normal StableSubgroup of H1; theorem [#]E is_stable_under_the_action_of A by Lm1; theorem [:O,{id E}:] is Action of O, E by Lm2; theorem for O being non empty set, E being set, o being Element of O, A being Action of O,E holds Product(<*o*>,A) = A.o by Lm26; theorem for O being non empty set, E being set, F1,F2 being FinSequence of O, A being Action of O,E holds Product(F1^F2,A) = Product(F1,A) * Product(F2,A) by Lm29; theorem for F being FinSequence of O, Y being Subset of E st Y is_stable_under_the_action_of A holds Product(F,A) .: Y c= Y by Lm30; theorem for E being non empty set, A being Action of O,E holds for X being Subset of E, a being Element of E st X is not empty holds a in the_stable_subset_generated_by(X,A) iff ex F being FinSequence of O, x being Element of X st Product(F,A).x = a by Lm31; theorem for G being strict Group holds ex H being strict GroupWithOperators of O st G = the multMagma of H proof let G be strict Group; consider H be non empty HGrWOpStr over O such that A1: H is strict distributive Group-like associative and A2: G = the multMagma of H by Lm3; reconsider H as strict GroupWithOperators of O by A1; take H; thus thesis by A2; end; theorem the multMagma of H1 is strict Subgroup of G by Lm16; theorem the multMagma of N is strict normal Subgroup of G by Lm7; theorem g1 in H1 implies (G^o).g1 in H1 by Lm10; theorem for O being set, G,H being GroupWithOperators of O, G9 being strict StableSubgroup of G, f being Homomorphism of G,H holds ex H9 being strict StableSubgroup of H st the carrier of H9 = f.:(the carrier of G9) by Lm17; theorem B is empty implies the_stable_subgroup_of B = (1).G by Lm25; theorem B = the carrier of gr C implies the_stable_subgroup_of C = the_stable_subgroup_of B by Lm32; theorem for N9 being normal Subgroup of G st N9 = the multMagma of N holds G ./.N9 = the multMagma of G./.N & 1_(G./.N9) = 1_(G./.N) by Lm35; theorem Th75: the carrier of H1 = the carrier of H2 implies the HGrWOpStr of H1 = the HGrWOpStr of H2 proof reconsider H19=H1,H29=H2 as Subgroup of G by Def7; A1: dom the action of H2 = O by FUNCT_2:def 1 .= dom the action of H1 by FUNCT_2:def 1; assume A2: the carrier of H1 = the carrier of H2; A3: now let x; assume A4: x in dom the action of H2; then reconsider o=x as Element of O; A5: H1^o = (the action of H1).o by A4,Def6; H1^o = (G^o)|the carrier of H2 by A2,Def7 .= H2^o by Def7; hence (the action of H1).x = (the action of H2).x by A4,A5,Def6; end; the multMagma of H19 = the multMagma of H29 by A2,GROUP_2:59; hence thesis by A1,A3,FUNCT_1:2; end; theorem Th76: H1./.N1 is trivial implies the HGrWOpStr of H1 = the HGrWOpStr of N1 proof reconsider N9=N1 as StableSubgroup of G by Th11; set H=H1; reconsider N=the multMagma of N1 as normal Subgroup of H by Lm7; assume A1: H1./.N1 is trivial; Cosets N1 = Cosets N by Def14; then consider e be set such that A2: the carrier of H./.N = {e} by A1,GROUP_6:def 2; A3: the carrier of H = union {e} by A2,GROUP_2:137; A4: now assume not the carrier of H c= the carrier of N; then (the carrier of H) \ (the carrier of N) <> {} by XBOOLE_1:37; then consider x such that A5: x in (the carrier of H) \ (the carrier of N) by XBOOLE_0:def 1; reconsider x as Element of H1 by A5; A6: now assume x * N = e; then x * N = the carrier of H by A3,ZFMISC_1:25; then consider x9 be Element of H such that A7: 1_H = x * x9 and A8: x9 in N by GROUP_2:103; x9=x" by A7,GROUP_1:12; then x"" in N by A8,GROUP_2:51; then x in carr(N) by STRUCT_0:def 5; hence contradiction by A5,XBOOLE_0:def 5; end; x * N in Cosets N by GROUP_6:14; hence contradiction by A2,A6,TARSKI:def 1; end; the carrier of N c= the carrier of H by GROUP_2:def 5; then the carrier of N9 = the carrier of H1 by A4,XBOOLE_0:def 10; hence thesis by Th75; end; theorem Th77: the carrier of H1 = the carrier of N1 implies H1./.N1 is trivial proof reconsider N19 = the multMagma of N1 as strict normal Subgroup of H1 by Lm7; assume A1: the carrier of H1 = the carrier of N1; now let x be set; hereby assume A2: x in Left_Cosets N19; then reconsider A=x as Subset of H1; consider a be Element of H1 such that A3: A = a * N19 by A2,GROUP_2:def 15; A = a * [#]the carrier of H1 by A1,A3; hence x = the carrier of H1 by GROUP_2:17; end; the carrier of H1 = 1_H1 * [#]the carrier of H1 by GROUP_2:17; then A4: the carrier of H1 = 1_H1 * N19 by A1; assume x = the carrier of H1; hence x in Left_Cosets N19 by A4,GROUP_2:def 15; end; then A5: {the carrier of H1} = Left_Cosets N19 by TARSKI:def 1; Cosets N1 = Cosets N19 by Def14; hence thesis by A5; end; :: ALG I.4.6 Proposition 7(a) theorem Th78: for G,H being GroupWithOperators of O, N being StableSubgroup of G, H9 being strict StableSubgroup of H, f being Homomorphism of G,H st N = Ker f holds ex G9 being strict StableSubgroup of G st the carrier of G9 = f"(the carrier of H9) & (H9 is normal implies N is normal StableSubgroup of G9 & G9 is normal) proof let G,H be GroupWithOperators of O; let N be StableSubgroup of G; let H9 be strict StableSubgroup of H; reconsider H99 = the multMagma of H9 as strict Subgroup of H by Lm16; let f be Homomorphism of G,H; assume A1: N = Ker f; set A = {g where g is Element of G:f.g in H99}; A2: 1_H in H99 by GROUP_2:46; then f.(1_G) in H99 by Lm13; then 1_G in A; then reconsider A as non empty set; now let x be set; assume x in A; then ex g be Element of G st x=g & f.g in H99; hence x in the carrier of G; end; then reconsider A as Subset of G by TARSKI:def 3; A3: now let g1,g2 be Element of G; assume that A4: g1 in A and A5: g2 in A; consider b be Element of G such that A6: b=g2 and A7: f.b in H99 by A5; consider a be Element of G such that A8: a=g1 and A9: f.a in H99 by A4; set fb = f.b; set fa = f.a; f.(a*b) = f.a * f.b & fa * fb in H99 by A9,A7,GROUP_2:50,GROUP_6:def 6; hence g1*g2 in A by A8,A6; end; A10: now let o be Element of O; let g be Element of G; assume g in A; then consider a be Element of G such that A11: a=g and A12: f.a in H99; f.a in the carrier of H99 by A12,STRUCT_0:def 5; then f.a in H9 by STRUCT_0:def 5; then (H^o).(f.g) in H9 by A11,Lm10; then f.((G^o).g) in H9 by Def18; then f.((G^o).g) in the carrier of H9 by STRUCT_0:def 5; then f.((G^o).g) in H99 by STRUCT_0:def 5; hence (G^o).g in A; end; now let g be Element of G; assume g in A; then consider a be Element of G such that A13: a=g and A14: f.a in H99; (f.a)" in H99 by A14,GROUP_2:51; then f.(a") in H99 by Lm14; hence g" in A by A13; end; then consider G99 be strict StableSubgroup of G such that A15: the carrier of G99 = A by A3,A10,Lm15; take G99; now reconsider R = f as Relation of the carrier of G, the carrier of H; let g be Element of G; hereby assume g in A; then ex a be Element of G st a=g & f.a in H99; then A16: f.g in the carrier of H9 by STRUCT_0:def 5; dom f = the carrier of G by FUNCT_2:def 1; then [g,f.g] in f by FUNCT_1:1; hence g in f"(the carrier of H9) by A16,RELSET_1:30; end; assume g in f"(the carrier of H9); then consider h be Element of H such that A17: [g,h] in R & h in (the carrier of H9) by RELSET_1:30; f.g=h & h in H99 by A17,FUNCT_1:1,STRUCT_0:def 5; hence g in A; end; hence the carrier of G99 = f"(the carrier of H9) by A15,SUBSET_1:3; reconsider G9 = the multMagma of G99 as strict Subgroup of G by Lm16; now assume A18: H9 is normal; now let g be Element of G; assume g in N; then f.g = 1_H by A1,Th47; then g in the carrier of G99 by A2,A15; hence g in G99 by STRUCT_0:def 5; end; hence N is normal StableSubgroup of G99 by A1,Th13,Th40; now let g be Element of G; now H99 is normal by A18,Def10; then A19: H99 |^ (f.g)" = H99 by GROUP_3:def 13; let x be set; assume x in g * G9; then consider h be Element of G such that A20: x=g*h and A21: h in A by A15,GROUP_2:27; set h9=g*h*g"; A22: f.h9 = f.(g*h) * f.(g") by GROUP_6:def 6 .= f.g * f.h * f.(g") by GROUP_6:def 6 .= f.g * f.h * (f.g)" by Lm14 .= ((f.g)")" * f.h * (f.g)" .= f.h |^ (f.g)" by GROUP_3:def 2; ex a be Element of G st a=h & f.a in H99 by A21; then f.h9 in H99 by A19,A22,GROUP_3:58; then A23: h9 in A; h9*g = (g*h)*(g"*g) by GROUP_1:def 3 .= (g*h)*1_G by GROUP_1:def 5 .= x by A20,GROUP_1:def 4; hence x in G9 * g by A15,A23,GROUP_2:28; end; hence g * G9 c= G9 * g by TARSKI:def 3; end; then for H being strict Subgroup of G st H = the multMagma of G99 holds H is normal by GROUP_3:118; hence G99 is normal by Def10; end; hence thesis; end; :: ALG I.4.6 Proposition 7(b) theorem Th79: for G,H being GroupWithOperators of O, N being StableSubgroup of G, G9 being strict StableSubgroup of G, f being Homomorphism of G,H st N = Ker f holds ex H9 being strict StableSubgroup of H st the carrier of H9 = f.:(the carrier of G9) & f"(the carrier of H9) = the carrier of G9"\/"N & (f is onto & G9 is normal implies H9 is normal) proof let G,H be GroupWithOperators of O; let N be StableSubgroup of G; reconsider N9=the multMagma of N as strict Subgroup of G by Lm16; let G9 be strict StableSubgroup of G; reconsider G99 = the multMagma of G9 as strict Subgroup of G by Lm16; let f be Homomorphism of G,H; set A = {f.g where g is Element of G:g in G99}; A1: G99*N9 = G9*N & N9*G99 = N*G9; 1_G in G99 by GROUP_2:46; then f.(1_G) in A; then reconsider A as non empty set; now let x be set; assume x in A; then ex g be Element of G st x=f.g & g in G99; hence x in the carrier of H; end; then reconsider A as Subset of H by TARSKI:def 3; A2: now let h1,h2 be Element of H; assume that A3: h1 in A and A4: h2 in A; consider a be Element of G such that A5: h1=f.a & a in G99 by A3; consider b be Element of G such that A6: h2=f.b & b in G99 by A4; f.(a*b) = h1*h2 & a*b in G99 by A5,A6,GROUP_2:50,GROUP_6:def 6; hence h1*h2 in A; end; A7: now let o be Element of O; let h be Element of H; assume h in A; then consider g be Element of G such that A8: h=f.g and A9: g in G99; g in the carrier of G99 by A9,STRUCT_0:def 5; then g in G9 by STRUCT_0:def 5; then (G^o).g in G9 by Lm10; then (G^o).g in the carrier of G9 by STRUCT_0:def 5; then A10: (G^o).g in G99 by STRUCT_0:def 5; (H^o).h = f.((G^o).g) by A8,Def18; hence (H^o).h in A by A10; end; now let h be Element of H; assume h in A; then consider g be Element of G such that A11: h=f.g & g in G99; g" in G99 & h"=f.(g") by A11,Lm14,GROUP_2:51; hence h" in A; end; then consider H99 be strict StableSubgroup of H such that A12: the carrier of H99 = A by A2,A7,Lm15; assume A13: N = Ker f; then N9 is normal by Def10; then A14: carr G99 * N9 = N9 * carr G99 by GROUP_3:120; reconsider H9 = the multMagma of H99 as strict Subgroup of H by Lm16; take H99; A15: now reconsider R = f as Relation of the carrier of G, the carrier of H; let h be Element of H; hereby assume h in A; then consider g be Element of G such that A16: h=f.g and A17: g in G99; A18: g in the carrier of G9 by A17,STRUCT_0:def 5; dom f = the carrier of G by FUNCT_2:def 1; then [g,h] in f by A16,FUNCT_1:1; hence h in f.:(the carrier of G9) by A18,RELSET_1:29; end; assume h in f.:(the carrier of G9); then consider g be Element of G such that A19: [g,h] in R & g in the carrier of G9 by RELSET_1:29; f.g=h & g in G99 by A19,FUNCT_1:1,STRUCT_0:def 5; hence h in A; end; hence A20: the carrier of H99 = f.:(the carrier of G9) by A12,SUBSET_1:3; A21: now let x; assume A22: x in f"(the carrier of H9); then f.x in the carrier of H9 by FUNCT_1:def 7; then consider g1 be set such that A23: g1 in dom f and A24: g1 in the carrier of G9 and A25: f.g1 = f.x by A20,FUNCT_1:def 6; reconsider g1,g2=x as Element of G by A22,A23; consider g3 be Element of G such that A26: g2 = g1 * g3 by GROUP_1:15; f.g2 = f.g2*f.g3 by A25,A26,GROUP_6:def 6; then f.g3 = 1_H by GROUP_1:7; then g3 in Ker f by Th47; then g3 in the carrier of N by A13,STRUCT_0:def 5; hence x in G99 * N9 by A24,A26; end; A27: dom f = the carrier of G by FUNCT_2:def 1; now let x; assume A28: x in G99 * N9; then consider g1,g2 be Element of G such that A29: x = g1*g2 and A30: g1 in carr G9 and A31: g2 in carr N9; A32: g2 in Ker f by A13,A31,STRUCT_0:def 5; f.x = f.g1*f.g2 by A29,GROUP_6:def 6 .= f.g1*1_H by A32,Th47 .= f.g1 by GROUP_1:def 4; then f.x in f.:(the carrier of G9) by A27,A30,FUNCT_1:def 6; then x in f"(f.:(the carrier of G9)) by A27,A28,FUNCT_1:def 7; hence x in f"(the carrier of H9) by A12,A15,SUBSET_1:3; end; then f"(the carrier of H9) = carr G9 * carr N by A21,TARSKI:1; hence f"(the carrier of H99) = the carrier of G9"\/"N by A14,A1,Th30; now assume that A33: f is onto and A34: G9 is normal; A35: G99 is normal by A34,Def10; now let h1 be Element of H; now let x be set; assume x in h1 * H9; then consider h2 be Element of H such that A36: x=h1*h2 and A37: h2 in A by A12,GROUP_2:27; set h29 = h1*h2*h1"; h2 in f.:(the carrier of G9) by A15,A37; then consider g2 be set such that A38: g2 in dom f and A39: g2 in the carrier of G99 and A40: f.g2 = h2 by FUNCT_1:def 6; rng f = the carrier of H by A33,FUNCT_2:def 3; then consider g1 be set such that A41: g1 in dom f and A42: h1 = f.g1 by FUNCT_1:def 3; reconsider g1,g2 as Element of G by A38,A41; set g29=g1*g2*g1"; g29=(g1"")*g2*g1"; then A43: g29=g2 |^ g1" by GROUP_3:def 2; g2 in G99 by A39,STRUCT_0:def 5; then g29 in G99 |^ g1" by A43,GROUP_3:58; then A44: g29 in the carrier of G99 |^ g1" by STRUCT_0:def 5; G99 |^ g1" is Subgroup of G99 by A35,GROUP_3:122; then A45: the carrier of G99 |^ g1" c= the carrier of G99 by GROUP_2:def 5; h29 = f.g1*f.g2*f.(g1") by A40,A42,Lm14 .= f.(g1*g2)*f.(g1") by GROUP_6:def 6 .= f.g29 by GROUP_6:def 6; then h29 in f.:(the carrier of G99) by A27,A44,A45,FUNCT_1:def 6; then A46: h29 in A by A15; h29*h1 = (h1*h2)*(h1"*h1) by GROUP_1:def 3 .= (h1*h2)*1_H by GROUP_1:def 5 .= x by A36,GROUP_1:def 4; hence x in H9 * h1 by A12,A46,GROUP_2:28; end; hence h1 * H9 c= H9 * h1 by TARSKI:def 3; end; then for H1 being strict Subgroup of H st H1 = the multMagma of H99 holds H1 is normal by GROUP_3:118; hence H99 is normal by Def10; end; hence thesis; end; theorem Th80: for G being strict GroupWithOperators of O, N being strict normal StableSubgroup of G, H being strict StableSubgroup of G./.N st the carrier of G = (nat_hom N)"(the carrier of H) holds H = (Omega).(G./.N) proof let G be strict GroupWithOperators of O; let N be strict normal StableSubgroup of G; reconsider N9 = the multMagma of N as strict normal Subgroup of G by Lm7; let H be strict StableSubgroup of G./.N; reconsider H9 = the multMagma of H as strict Subgroup of G./.N by Lm16; A1: the carrier of H9 c= the carrier of G./.N & the multF of H9 = (the multF of G./.N)||the carrier of H9 by GROUP_2:def 5; the carrier of G./.N = the carrier of G./.N9 & the multF of G./.N = the multF of G./.N9 by Def14,Def15; then reconsider H9 as strict Subgroup of G./.N9 by A1,GROUP_2:def 5; assume the carrier of G = (nat_hom N)"(the carrier of H); then A2: the carrier of G = (nat_hom N9)"(the carrier of H9) by Def20; now reconsider R = nat_hom N9 as Relation of the carrier of G, the carrier of G./.N9; let h be Element of G./.N9; thus h in H9 implies h in (Omega).(G./.N9) by STRUCT_0:def 5; assume h in (Omega).(G./.N9); h in Left_Cosets N9; then consider g be Element of G such that A3: h = g * N9 by GROUP_2:def 15; consider h9 be Element of G./.N9 such that A4: [g,h9] in R and A5: h9 in (the carrier of H9) by A2,RELSET_1:30; (nat_hom N9).g = h9 by A4,FUNCT_1:1; then h in the carrier of H9 by A3,A5,GROUP_6:def 8; hence h in H9 by STRUCT_0:def 5; end; then H9 = (Omega).(G./.N9) by GROUP_2:def 6; then the carrier of H = Cosets N by Def14; hence thesis by Lm5; end; theorem Th81: for G being strict GroupWithOperators of O, N being strict normal StableSubgroup of G, H being strict StableSubgroup of G./.N st the carrier of N = (nat_hom N)"(the carrier of H) holds H = (1).(G./.N) proof let G be strict GroupWithOperators of O; let N be strict normal StableSubgroup of G; reconsider N9 = the multMagma of N as strict normal Subgroup of G by Lm7; let H be strict StableSubgroup of G./.N; reconsider H9 = the multMagma of H as strict Subgroup of G./.N by Lm16; A1: the carrier of H9 c= the carrier of G./.N & the multF of H9 = (the multF of G./.N)||the carrier of H9 by GROUP_2:def 5; the carrier of G./.N = the carrier of G./.N9 & the multF of G./.N = the multF of G./.N9 by Def14,Def15; then reconsider H9 as strict Subgroup of G./.N9 by A1,GROUP_2:def 5; assume the carrier of N = (nat_hom N)"(the carrier of H); then A2: the carrier of N9 = (nat_hom N9)"(the carrier of H9) by Def20; assume not H = (1).(G./.N); then not the carrier of H = {1_(G./.N)} by Def8; then consider h be set such that A3: not (h in the carrier of H iff h in {1_(G./.N)}) by TARSKI:1; per cases by A3; suppose A4: h in the carrier of H & not h in {1_(G./.N)}; then {h} c= the carrier of H by ZFMISC_1:31; then A5: (nat_hom N9)"{h} c= the carrier of N9 by A2,RELAT_1:143; A6: rng nat_hom N9 = the carrier of Image nat_hom N9 by GROUP_6:44 .= the carrier of G./.N9 by GROUP_6:48; the carrier of H9 c= the carrier of G./.N9 by GROUP_2:def 5; then consider x be set such that A7: x in dom nat_hom N9 and A8: (nat_hom N9).x = h by A4,A6,FUNCT_1:def 3; (nat_hom N9).x in {h} by A8,TARSKI:def 1; then x in (nat_hom N9)"{h} by A7,FUNCT_1:def 7; then A9: x in N9 by A5,STRUCT_0:def 5; h <> 1_(G./.N) by A4,TARSKI:def 1; then A10: h <> carr N by Th43; reconsider x as Element of G by A7; x * N9 = h by A8,GROUP_6:def 8; hence contradiction by A10,A9,GROUP_2:113; end; suppose not h in the carrier of H & h in {1_(G./.N)}; then h = 1_(G./.N) & not h in H by STRUCT_0:def 5,TARSKI:def 1; hence contradiction by Lm18; end; end; theorem Th82: for G,H being strict GroupWithOperators of O st G,H are_isomorphic & G is simple holds H is simple proof let G,H be strict GroupWithOperators of O; assume A1: G,H are_isomorphic; assume A2: G is simple; assume A3: H is not simple; per cases by A3,Def13; suppose H is trivial; then G is trivial by A1,Th58; hence contradiction by A2,Def13; end; suppose ex H9 being strict normal StableSubgroup of H st H9 <> (Omega).H & H9 <> (1).H; then consider H9 be strict normal StableSubgroup of H such that A4: H9 <> (Omega).H and A5: H9 <> (1).H; consider f be Homomorphism of G,H such that A6: f is bijective by A1,Def19; reconsider H99 = the multMagma of H9 as strict normal Subgroup of H by Lm7; the multMagma of H9 <> the multMagma of H by A4,Lm5; then consider h be Element of H such that A7: not h in H99 by GROUP_2:62; the carrier of H9<>{1_H} by A5,Def8; then consider x be set such that A8: x in the carrier of H9 and A9: x<>1_H by ZFMISC_1:35; A10: x in H99 by A8,STRUCT_0:def 5; then x in H by GROUP_2:40; then reconsider x as Element of H by STRUCT_0:def 5; consider y be Element of G such that A11: f.y = x by A6,Th52; set A = {g where g is Element of G: f.g in H99}; consider g be Element of G such that A12: f.g = h by A6,Th52; 1_H in H99 by GROUP_2:46; then f.(1_G) in H99 by Lm13; then 1_G in A; then reconsider A as non empty set; now let x be set; assume x in A; then ex g be Element of G st x=g & f.g in H99; hence x in the carrier of G; end; then reconsider A as Subset of G by TARSKI:def 3; A13: now let g1,g2 be Element of G; assume that A14: g1 in A and A15: g2 in A; consider b be Element of G such that A16: b=g2 and A17: f.b in H99 by A15; consider a be Element of G such that A18: a=g1 and A19: f.a in H99 by A14; set fb = f.b; set fa = f.a; f.(a*b) = f.a * f.b & fa * fb in H99 by A19,A17,GROUP_2:50,GROUP_6:def 6; hence g1*g2 in A by A18,A16; end; A20: now let o be Element of O; let g be Element of G; assume g in A; then consider a be Element of G such that A21: a=g and A22: f.a in H99; f.a in the carrier of H99 by A22,STRUCT_0:def 5; then f.a in H9 by STRUCT_0:def 5; then (H^o).(f.g) in H9 by A21,Lm10; then f.((G^o).g) in H9 by Def18; then f.((G^o).g) in the carrier of H9 by STRUCT_0:def 5; then f.((G^o).g) in H99 by STRUCT_0:def 5; hence (G^o).g in A; end; now let g be Element of G; assume g in A; then consider a be Element of G such that A23: a=g and A24: f.a in H99; (f.a)" in H99 by A24,GROUP_2:51; then f.(a") in H99 by Lm14; hence g" in A by A23; end; then consider G99 be strict StableSubgroup of G such that A25: the carrier of G99 = A by A13,A20,Lm15; reconsider G9=the multMagma of G99 as strict Subgroup of G by Lm16; now let g be Element of G; now let x be set; A26: H99 |^ (f.g)" = H99 by GROUP_3:def 13; assume x in g * G9; then consider h be Element of G such that A27: x=g*h and A28: h in A by A25,GROUP_2:27; set h9=g*h*g"; A29: f.h9 = f.(g*h) * f.(g") by GROUP_6:def 6 .= f.g * f.h * f.(g") by GROUP_6:def 6 .= f.g * f.h * (f.g)" by Lm14 .= ((f.g)")" * f.h * (f.g)" .= f.h |^ (f.g)" by GROUP_3:def 2; ex a be Element of G st a=h & f.a in H99 by A28; then f.h9 in H99 by A26,A29,GROUP_3:58; then A30: h9 in A; h9*g = (g*h)*(g"*g) by GROUP_1:def 3 .= (g*h)*1_G by GROUP_1:def 5 .= x by A27,GROUP_1:def 4; hence x in G9 * g by A25,A30,GROUP_2:28; end; hence g * G9 c= G9 * g by TARSKI:def 3; end; then for H being strict Subgroup of G st H = the multMagma of G99 holds H is normal by GROUP_3:118; then A31: G99 is normal by Def10; A32: y<>1_G by A9,A11,Lm13; y in the carrier of G99 by A25,A10,A11; then the carrier of G99 <> {1_G} by A32,TARSKI:def 1; then A33: G99<>(1).G by Def8; now assume g in A; then ex g9 be Element of G st g9=g & f.g9 in H99; hence contradiction by A7,A12; end; then G99<>(Omega).G by A25; hence contradiction by A2,A33,A31,Def13; end; end; theorem Th83: for G being GroupWithOperators of O, H being StableSubgroup of G , FG being FinSequence of the carrier of G, FH being FinSequence of the carrier of H, I be FinSequence of INT st FG=FH & len FG = len I holds Product(FG |^ I) = Product(FH |^ I) proof let G be GroupWithOperators of O; let H be StableSubgroup of G; let FG be FinSequence of the carrier of G; let FH be FinSequence of the carrier of H; let I be FinSequence of INT; assume A1: FG=FH & len FG = len I; defpred P[Nat] means for FG being FinSequence of the carrier of G, FH being FinSequence of the carrier of H, I being FinSequence of INT st len FG = $1 & FG =FH & len FG = len I holds Product(FG |^ I) = Product(FH |^ I); A2: now let n be Nat; assume A3: P[n]; thus P[n+1] proof let FG be FinSequence of the carrier of G; let FH be FinSequence of the carrier of H; let I be FinSequence of INT; assume A4: len FG = n+1; then consider FGn be FinSequence of the carrier of G, g be Element of G such that A5: FG = FGn^<*g*> by FINSEQ_2:19; A6: len FG = len FGn + len <*g*> by A5,FINSEQ_1:22; then A7: n+1 = len FGn + 1 by A4,FINSEQ_1:40; assume that A8: FG=FH and A9: len FG = len I; consider FHn be FinSequence of the carrier of H, h be Element of H such that A10: FH = FHn^<*h*> by A4,A8,FINSEQ_2:19; consider In be FinSequence of INT, i be Element of INT such that A11: I = In^<*i*> by A4,A9,FINSEQ_2:19; set FG1=<*g*>; set I1=<*i*>; len I = len In + len <*i*> by A11,FINSEQ_1:22; then A12: n+1 = len In + 1 by A4,A9,FINSEQ_1:40; A13: len FH = len FHn + len <*h*> by A10,FINSEQ_1:22; then A14: FH.(n+1)=(FHn^<*h*>).(len FHn +1) by A4,A8,A10,FINSEQ_1:40 .= h by FINSEQ_1:42; A15: n+1 = len FHn + 1 by A4,A8,A13,FINSEQ_1:40; A16: FG.(n+1)=(FGn^<*g*>).(len FGn +1) by A4,A5,A6,FINSEQ_1:40 .= g by FINSEQ_1:42; A17: now reconsider H9=H as Subgroup of G by Def7; reconsider h9=h as Element of H9; g|^i = h9|^i by A8,A16,A14,GROUP_4:2; hence g|^i = h|^i; end; len FG1 = 1 by FINSEQ_1:40 .=len I1 by FINSEQ_1:40; then A18: Product(FG |^ I) = Product((FGn |^ In)^(FG1 |^ I1)) by A11,A5,A12,A7, GROUP_4:19 .= Product(FGn |^ In) * Product(FG1 |^ I1) by GROUP_4:5; set FH1=<*h*>; A19: len FH1 = 1 by FINSEQ_1:40 .=len I1 by FINSEQ_1:40; A20: Product(FG1 |^ I1) = Product(<*g*>|^<*@i*>) .= Product <*g|^i*> by GROUP_4:22 .= h|^i by A17,GROUP_4:9 .= Product <*h|^i*> by GROUP_4:9 .= Product(<*h*>|^<*@i*>) by GROUP_4:22 .= Product(FH1 |^ I1); FGn = FHn by A8,A5,A10,A16,A14,FINSEQ_1:33; then Product(FGn |^ In) = Product(FHn |^ In) by A3,A12,A15; then Product(FG |^ I) = Product(FHn |^ In) * Product(FH1 |^ I1) by A18 ,A20,Th3 .= Product((FHn |^ In)^(FH1 |^ I1)) by GROUP_4:5 .= Product((FHn^FH1) |^ (In^I1)) by A12,A15,A19,GROUP_4:19; hence thesis by A11,A10; end; end; A21: P[0] proof let FG be FinSequence of the carrier of G; let FH be FinSequence of the carrier of H; let I be FinSequence of INT; assume A22: len FG = 0; then len(FG |^ I) = 0 by GROUP_4:def 3; then FG |^ I = <*> the carrier of G; then A23: Product(FG |^ I) = 1_G by GROUP_4:8; assume that A24: FG=FH and len FG = len I; len(FH |^ I) = 0 by A22,A24,GROUP_4:def 3; then FH |^ I = <*> the carrier of H; then Product(FH |^ I) = 1_H by GROUP_4:8; hence thesis by A23,Th4; end; for n being Nat holds P[n] from NAT_1:sch 2(A21,A2); hence thesis by A1; end; theorem Th84: for O,E1,E2 being set, A1 being Action of O,E1, A2 being Action of O,E2, F being FinSequence of O st E1 c= E2 & (for o being Element of O, f1 being Function of E1,E1, f2 being Function of E2,E2 st f1=A1.o & f2=A2.o holds f1 = f2|E1) holds Product(F,A1) = Product(F,A2)|E1 proof let O,E1,E2 be set; let A1 be Action of O,E1; let A2 be Action of O,E2; let F be FinSequence of O; defpred P[Element of NAT] means for F being FinSequence of O st len F = $1 holds Product(F,A1) = Product(F,A2)|E1; assume A1: E1 c= E2; A2: P[0] proof let F be FinSequence of O; A3: now let x; assume A4: x in dom id E1; then A5: x in E1; thus (id E1).x = x by A4,FUNCT_1:18 .= (id E2).x by A1,A5,FUNCT_1:18; end; E1 = E2 /\ E1 by A1,XBOOLE_1:28; then dom id E1 = E2 /\ E1; then A6: dom id E1 = dom id E2 /\ E1; assume A7: len F = 0; hence Product(F,A1) = id E1 by Def3 .= (id E2)|E1 by A6,A3,FUNCT_1:46 .= Product(F,A2)|E1 by A7,Def3; end; assume A8: for o being Element of O, f1 being Function of E1,E1, f2 being Function of E2,E2 st f1=A1.o & f2=A2.o holds f1 = f2|E1; per cases; suppose O is empty; then len F = 0; hence thesis by A2; end; suppose A9: O is non empty; A10: for k being Element of NAT st P[k] holds P[k + 1] proof let k be Element of NAT; assume A11: P[k]; now let F be FinSequence of O; assume A12: len F = k+1; then consider Fk be FinSequence of O, o be Element of O such that A13: F = Fk^<*o*> by FINSEQ_2:19; len F = len Fk + len <*o*> by A13,FINSEQ_1:22; then A14: k+1 = len Fk + 1 by A12,FINSEQ_1:39; A15: now {o} c= O by A9,ZFMISC_1:31; then rng <*o*> c= O by FINSEQ_1:38; then reconsider Fo=<*o*> as FinSequence of O by FINSEQ_1:def 4; let x; assume A16: x in dom Product(F,A1); then A17: x in E1; A18: o in O by A9; then o in dom A1 by FUNCT_2:def 1; then A1.o in rng A1 by FUNCT_1:3; then consider f1 be Function such that A19: f1=A1.o and A20: dom f1 = E1 and A21: rng f1 c= E1 by FUNCT_2:def 2; A22: Product(Fo,A1) = f1 by A9,A19,Lm26; o in dom A2 by A18,FUNCT_2:def 1; then A2.o in rng A2 by FUNCT_1:3; then consider f2 be Function such that A23: f2=A2.o and A24: dom f2 = E2 and rng f2 c= E2 by FUNCT_2:def 2; A25: Product(Fo,A2) = f2 by A9,A23,Lm26; A26: f1.x in rng f1 by A16,A20,FUNCT_1:3; A27: Product(F,A2) = (Product(Fk,A2)*Product(Fo,A2)) by A9,A13,Lm29 .= Product(Fk,A2)*f2 by A9,A23,Lm26; Product(F,A1) = (Product(Fk,A1)*Product(Fo,A1)) by A9,A13,Lm29 .= Product(Fk,A1)*f1 by A9,A19,Lm26; hence Product(F,A1).x = Product(Fk,A1).(f1.x) by A16,A20,FUNCT_1:13 .= (Product(Fk,A2)|E1).(f1.x) by A11,A14 .= Product(Fk,A2).(f1.x) by A21,A26,FUNCT_1:49 .= Product(Fk,A2).((f2|E1).x) by A8,A19,A23,A22,A25 .= Product(Fk,A2).(f2.x) by A16,FUNCT_1:49 .= (Product(Fk,A2)*f2).x by A1,A17,A24,FUNCT_1:13 .= (Product(F,A2)|E1).x by A16,A27,FUNCT_1:49; end; Product(F,A2) in Funcs(E2,E2) by FUNCT_2:9; then ex f2 be Function st Product(F,A2) = f2 & dom f2 = E2 & rng f2 c= E2 by FUNCT_2:def 2; then A28: dom(Product(F,A2)|E1) = E2 /\ E1 by RELAT_1:61 .= E1 by A1,XBOOLE_1:28; Product(F,A1) in Funcs(E1,E1) by FUNCT_2:9; then ex f1 be Function st Product(F,A1) = f1 & dom f1 = E1 & rng f1 c= E1 by FUNCT_2:def 2; hence Product(F,A1) = Product(F,A2)|E1 by A28,A15,FUNCT_1:2; end; hence thesis; end; A29: for k being Element of NAT holds P[k] from NAT_1:sch 1(A2,A10); reconsider k = len F as Element of NAT; k = len F; hence thesis by A29; end; end; theorem Th85: for N1,N2 being strict StableSubgroup of H1, N19,N29 being strict StableSubgroup of G st N1 = N19 & N2 = N29 holds N19 * N29 = N1 * N2 proof let N1,N2 be strict StableSubgroup of H1; let N19,N29 be strict StableSubgroup of G; set X={g * h where g,h is Element of G: g in carr N19 & h in carr N29}; set Y={g * h where g,h is Element of H1: g in carr N1 & h in carr N2}; assume A1: N1=N19 & N2=N29; A2: now N2 is Subgroup of H1 by Def7; then A3: the carrier of N2 c= the carrier of H1 by GROUP_2:def 5; let x; assume x in X; then consider g,h be Element of G such that A4: x=g*h and A5: g in carr N19 & h in carr N29; N1 is Subgroup of H1 by Def7; then the carrier of N1 c= the carrier of H1 by GROUP_2:def 5; then reconsider g,h as Element of H1 by A1,A5,A3; x=g*h by A4,Th3; hence x in Y by A1,A5; end; now let x; assume x in Y; then consider g,h be Element of H1 such that A6: x=g*h and A7: g in carr N1 & h in carr N2; reconsider g,h as Element of G by Th2; x=g*h by A6,Th3; hence x in X by A1,A7; end; hence thesis by A2,TARSKI:1; end; theorem Th86: for N1,N2 being strict StableSubgroup of H1, N19,N29 being strict StableSubgroup of G st N1 = N19 & N2 = N29 holds N19 "\/" N29 = N1 "\/" N2 proof let N1,N2 be strict StableSubgroup of H1; reconsider S2 = the_stable_subgroup_of(N1*N2) as StableSubgroup of G by Th11; let N19,N29 be strict StableSubgroup of G; set S1 = the_stable_subgroup_of(N19*N29); set X1={B where B is Subset of G : ex H being strict StableSubgroup of G st B = the carrier of H & N19*N29 c= carr H}; set X2={B where B is Subset of H1 : ex H being strict StableSubgroup of H1 st B = the carrier of H & N1*N2 c= carr H}; A1: N19 "\/" N29 = the_stable_subgroup_of(N19*N29) & N1 "\/" N2 = the_stable_subgroup_of(N1*N2) by Th29; A2: the carrier of the_stable_subgroup_of(N19*N29) = meet X1 & the carrier of the_stable_subgroup_of(N1*N2) = meet X2 by Th27; assume A3: N1=N19 & N2=N29; now let x; assume x in X2; then consider B be Subset of H1 such that A4: x=B and A5: ex H being strict StableSubgroup of H1 st B = the carrier of H & N1 *N2 c= carr H; now consider H be strict StableSubgroup of H1 such that A6: B = the carrier of H & N1*N2 c= carr H by A5; reconsider H as strict StableSubgroup of G by Th11; take H; thus B = the carrier of H & N19*N29 c= carr H by A3,A6,Th85; end; hence x in X1 by A4; end; then A7: X2 c= X1 by TARSKI:def 3; now set x9=carr H1; reconsider x=x9 as set; take x; now set H=(Omega).H1; take H; thus x9 = the carrier of H; thus N1*N2 c= carr H; end; hence x in X2; end; then A8: meet X1 c= meet X2 by A7,SETFAM_1:6; now let x; assume A9: x in the carrier of the_stable_subgroup_of(N1*N2); the_stable_subgroup_of(N1*N2) is Subgroup of H1 by Def7; then the carrier of the_stable_subgroup_of(N1*N2) c= the carrier of H1 by GROUP_2:def 5; then reconsider g=x as Element of H1 by A9; g in the_stable_subgroup_of(N1*N2) by A9,STRUCT_0:def 5; then consider F be FinSequence of the carrier of H1, I be FinSequence of INT, C be Subset of H1 such that A10: C = the_stable_subset_generated_by (N1*N2, the action of H1) and A11: len F = len I and A12: rng F c= C and A13: Product(F |^ I) = g by Th24; now N2 is Subgroup of H1 by Def7; then 1_H1 in N2 by GROUP_2:46; then A14: 1_H1 in carr N2 by STRUCT_0:def 5; let x; assume A15: x in the_stable_subset_generated_by (N1*N2, the action of H1); then reconsider a=x as Element of H1; N1 is Subgroup of H1 by Def7; then 1_H1 in N1 by GROUP_2:46; then A16: 1_H1 in carr N1 by STRUCT_0:def 5; 1_H1=1_H1*1_H1 by GROUP_1:def 4; then A17: 1_H1 in carr N1 * carr N2 by A16,A14; then consider F be FinSequence of O, h be Element of N1*N2 such that A18: Product(F,the action of H1).h = a by A15,Lm31; H1 is Subgroup of G by Def7; then A19: the carrier of H1 c= the carrier of G by GROUP_2:def 5; then reconsider a as Element of G by TARSKI:def 3; A20: h in N1*N2 by A17; reconsider h as Element of N19*N29 by A3,Th85; now let o be Element of O; let f1 be Function of the carrier of H1,the carrier of H1; let f2 be Function of the carrier of G,the carrier of G; assume that A21: f1=(the action of H1).o and A22: f2=(the action of G).o; per cases; suppose o in O; then H1^o = f1 & G^o = f2 by A21,A22,Def6; hence f1 = f2|the carrier of H1 by Def7; end; suppose not o in O; then not o in dom the action of H1; hence f1 = f2|the carrier of H1 by A21,FUNCT_1:def 2; end; end; then Product(F,the action of H1) = Product(F,the action of G)|the carrier of H1 by A19,Th84; then A23: Product(F,the action of G).h = a by A18,A20,FUNCT_1:49; N19*N29 is non empty by A3,A20,Th85; hence x in the_stable_subset_generated_by (N19*N29, the action of G) by A23,Lm31; end; then the_stable_subset_generated_by (N1*N2, the action of H1) c= the_stable_subset_generated_by (N19*N29, the action of G) by TARSKI:def 3; then A24: rng F c= the_stable_subset_generated_by (N19*N29, the action of G ) by A10,A12,XBOOLE_1:1; reconsider g as Element of G by Th2; H1 is Subgroup of G by Def7; then the carrier of H1 c= the carrier of G by GROUP_2:def 5; then rng F c= the carrier of G by XBOOLE_1:1; then reconsider F as FinSequence of the carrier of G by FINSEQ_1:def 4; Product(F |^ I) = g by A11,A13,Th83; then A25: g in the_stable_subgroup_of(N19*N29) by A11,A24,Th24; assume not x in the carrier of the_stable_subgroup_of(N19*N29); hence contradiction by A25,STRUCT_0:def 5; end; then meet X2 c= meet X1 by A2,TARSKI:def 3; then the carrier of S1 = the carrier of S2 by A2,A8,XBOOLE_0:def 10; hence thesis by A1,Lm5; end; theorem Th87: for N1,N2 being strict StableSubgroup of G st N1 is normal StableSubgroup of H1 & N2 is normal StableSubgroup of H1 holds N1 "\/" N2 is normal StableSubgroup of H1 proof let N1,N2 be strict StableSubgroup of G; assume A1: N1 is normal StableSubgroup of H1 & N2 is normal StableSubgroup of H1; then reconsider N19=N1,N29=N2 as StableSubgroup of H1; N1 "\/" N2 = N19 "\/" N29 by Th86; hence thesis by A1,Th32; end; theorem Th88: for f being Homomorphism of G,H holds for g being Homomorphism of H,I holds the carrier of Ker(g*f) = f"(the carrier of Ker g) proof let f be Homomorphism of G,H; let g be Homomorphism of H,I; A1: now let x; assume A2: x in f"(the carrier of Ker g); then f.x in the carrier of Ker g by FUNCT_1:def 7; then f.x in {b where b is Element of H: g.b = 1_I} by Def21; then A3: ex b be Element of H st b=f.x & g.b = 1_I; x in dom f by A2,FUNCT_1:def 7; then 1_I = (g*f).x by A3,FUNCT_1:13; then x in {a9 where a9 is Element of G: (g*f).a9 = 1_I} by A2; hence x in the carrier of Ker(g*f) by Def21; end; A4: dom f = the carrier of G by FUNCT_2:def 1; now let x; assume x in the carrier of Ker(g*f); then x in {a where a is Element of G: (g*f).a = 1_I} by Def21; then consider a be Element of G such that A5: x=a and A6: (g*f).a =1_I; reconsider b=f.a as Element of H; g.b = 1_I by A4,A6,FUNCT_1:13; then f.x in {b9 where b9 is Element of H: g.b9 = 1_I} by A5; then f.x in the carrier of Ker g by Def21; hence x in f"(the carrier of Ker g) by A4,A5,FUNCT_1:def 7; end; hence thesis by A1,TARSKI:1; end; theorem Th89: for G9 being StableSubgroup of G, H9 being StableSubgroup of H, f being Homomorphism of G,H st the carrier of H9 = f.:(the carrier of G9) or the carrier of G9 = f"(the carrier of H9) holds f|(the carrier of G9) is Homomorphism of G9,H9 proof let G9 be StableSubgroup of G; let H9 be StableSubgroup of H; let f be Homomorphism of G,H; set g=f|(the carrier of G9); G9 is Subgroup of G by Def7; then A1: the carrier of G9 c= the carrier of G by GROUP_2:def 5; then A2: the carrier of G9 c= dom f by FUNCT_2:def 1; then A3: dom g = the carrier of G9 by RELAT_1:62; assume A4: the carrier of H9 = f.:(the carrier of G9) or the carrier of G9 = f" (the carrier of H9); A5: for x st x in the carrier of G9 holds f.x in the carrier of H9 proof let x; assume A6: x in the carrier of G9; per cases by A4; suppose A7: the carrier of H9 = f.:(the carrier of G9); assume not f.x in the carrier of H9; hence contradiction by A2,A6,A7,FUNCT_1:def 6; end; suppose the carrier of G9 = f"(the carrier of H9); hence thesis by A6,FUNCT_1:def 7; end; end; now let y be set; assume y in rng g; then consider x such that A8: x in dom g and A9: y = g.x by FUNCT_1:def 3; A10: x in the carrier of G9 by A2,A8,RELAT_1:62; then y = f.x by A9,FUNCT_1:49; hence y in the carrier of H9 by A5,A10; end; then rng g c= the carrier of H9 by TARSKI:def 3; then reconsider g as Function of G9,H9 by A3,RELSET_1:4; A11: now let a9,b9 be Element of G9; reconsider a=a9,b=b9 as Element of G by A1,TARSKI:def 3; A12: f.a = g.a9 & f.b = g.b9 by FUNCT_1:49; thus g.(a9* b9) = f.(a9*b9) by FUNCT_1:49 .= f.(a*b) by Th3 .= f.a * f.b by GROUP_6:def 6 .= g.a9 * g.b9 by A12,Th3; end; now let o be Element of O; let a9 be Element of G9; reconsider a=a9 as Element of G by A1,TARSKI:def 3; thus g.((G9^o).a9) = f.((G9^o).a9) by FUNCT_1:49 .= f.(((G^o)|the carrier of G9).a9) by Def7 .= f.((G^o).a) by FUNCT_1:49 .= (H^o).(f.a) by Def18 .= (H^o).(g.a9) by FUNCT_1:49 .= ((H^o)|the carrier of H9).(g.a9) by FUNCT_1:49 .= (H9^o).(g.a9) by Def7; end; hence thesis by A11,Def18,GROUP_6:def 6; end; :: ALG I.4.6 Corollary 2 theorem Th90: for G,H being strict GroupWithOperators of O, N,L,G9 being strict StableSubgroup of G, f being Homomorphism of G,H st N = Ker f & L is strict normal StableSubgroup of G9 holds L"\/"(G9/\N) is normal StableSubgroup of G9 & L"\/"N is normal StableSubgroup of G9"\/"N & for N1 being strict normal StableSubgroup of G9"\/"N, N2 being strict normal StableSubgroup of G9 st N1=L "\/"N & N2=L"\/"(G9/\N) holds (G9"\/"N)./.N1, G9./.N2 are_isomorphic proof let G,H be strict GroupWithOperators of O; let N,L,G9 be strict StableSubgroup of G; reconsider N9=G9/\N as StableSubgroup of G9 by Lm34; reconsider Gs9 = the multMagma of G9 as strict Subgroup of G by Lm16; let f be Homomorphism of G,H; reconsider L99=L as Subgroup of G by Def7; assume A1: N = Ker f; then consider H9 be strict StableSubgroup of H such that A2: the carrier of H9 = f.:(the carrier of G9) and A3: f"(the carrier of H9) = the carrier of G9"\/"N and f is onto & G9 is normal implies H9 is normal by Th79; reconsider f99 = f|(the carrier of G9"\/"N) as Homomorphism of G9"\/"N,H9 by A3,Th89; reconsider Ns = the multMagma of N as strict normal Subgroup of G by A1,Lm7; carr Gs9 * Ns = Ns * carr Gs9 by GROUP_3:120; then A4: G9 * N = N * G9; A5: now let y be set; assume y in f.:(the carrier of G9); then consider x such that A6: x in dom f and A7: x in the carrier of G9 and A8: y = f.x by FUNCT_1:def 6; reconsider x as Element of G by A6; consider x9 be set such that A9: x9=x*1_G; A10: x9 in dom f by A6,A9,GROUP_1:def 4; A11: y = f.x * 1_H by A8,GROUP_1:def 4 .= f.x * f.(1_G) by Lm13 .= f.x9 by A9,GROUP_6:def 6; f.(1_G) = 1_H by Lm13; then 1_G in Ker f by Th47; then 1_G in carr N by A1,STRUCT_0:def 5; then x9 in G9*N by A7,A9; hence y in f.:(G9*N) by A10,A11,FUNCT_1:def 6; end; A12: dom f = the carrier of G by FUNCT_2:def 1; now let y be set; assume y in f.:(G9*N); then consider x such that A13: x in dom f and A14: x in G9*N and A15: y = f.x by FUNCT_1:def 6; reconsider x as Element of G by A13; consider g1,g2 be Element of G such that A16: x = g1*g2 and A17: g1 in carr G9 and A18: g2 in carr N by A14; A19: g2 in N by A18,STRUCT_0:def 5; y = f.g1*f.g2 by A15,A16,GROUP_6:def 6 .= f.g1*1_H by A1,A19,Th47 .= f.g1 by GROUP_1:def 4; hence y in f.:(the carrier of G9) by A12,A17,FUNCT_1:def 6; end; then f.:(the carrier of G9) = f.:(G9*N) by A5,TARSKI:1; then A20: f99.:(the carrier of (G9"\/"N))=f.:(the carrier of (G9"\/"N)) & the carrier of H9 = f.:(the carrier of (G9"\/"N)) by A2,A4,Th30,RELAT_1:129; A21: now let x; assume x in f99"(f.:(the carrier of L)); then A22: x in ((the carrier of G9"\/"N) /\ f"(f.:(the carrier of L))) by FUNCT_1:70 ; then x in f"(f.:(the carrier of L)) by XBOOLE_0:def 4; then f.x in f.:(the carrier of L) by FUNCT_1:def 7; then consider g1 be set such that A23: g1 in dom f and A24: g1 in the carrier of L and A25: f.x = f.g1 by FUNCT_1:def 6; reconsider g1,g2=x as Element of G by A22,A23; consider g3 be Element of G such that A26: g2 = g1 * g3 by GROUP_1:15; f.g2 = f.g2*f.g3 by A25,A26,GROUP_6:def 6; then f.g3 = 1_H by GROUP_1:7; then g3 in Ker f by Th47; then g3 in the carrier of N by A1,STRUCT_0:def 5; hence x in L * N by A24,A26; end; reconsider f9=f|(the carrier of G9) as Homomorphism of G9,H9 by A2,Th89; A27: now let x; assume x in the carrier of N9; then A28: x in carr G9 /\ carr N by Def25; then reconsider a9=x as Element of G9 by XBOOLE_0:def 4; reconsider a99=a9 as Element of G by Th2; x in carr N by A28,XBOOLE_0:def 4; then x in N by STRUCT_0:def 5; then f.a99 = 1_H by A1,Th47; then f.a9 = 1_H9 by Th4; then f9.a9 = 1_H9 by FUNCT_1:49; hence x in {a where a is Element of G9: f9.a = 1_H9}; end; assume A29: L is strict normal StableSubgroup of G9; then reconsider L9=L as strict StableSubgroup of G9; reconsider N1=L"\/"N as StableSubgroup of G9"\/"N by A29,Th38; carr L99 * Ns = Ns * carr L99 by GROUP_3:120; then A30: L * N = N * L; now let x; assume x in {a where a is Element of G9: f9.a = 1_H9}; then consider a be Element of G9 such that A31: x=a and A32: f9.a = 1_H9; reconsider a as Element of G by Th2; f.a = 1_H9 by A32,FUNCT_1:49; then f.a = 1_H by Th4; then x in N by A1,A31,Th47; then x in carr N by STRUCT_0:def 5; then x in carr G9 /\ carr N by A31,XBOOLE_0:def 4; hence x in the carrier of N9 by Def25; end; then the carrier of N9 = {a where a is Element of G9: f9.a = 1_H9} by A27,TARSKI:1; then A33: N9 = Ker f9 by Def21; then consider H99 be strict StableSubgroup of H9 such that A34: the carrier of H99 = f9.:(the carrier of L9) and A35: f9"(the carrier of H99) = the carrier of L9"\/"N9 and A36: f9 is onto & L9 is normal implies H99 is normal by Th79; consider N2 be strict StableSubgroup of G9 such that A37: the carrier of N2 = f9"(the carrier of H99) and A38: H99 is normal implies N9 is normal StableSubgroup of N2 & N2 is normal by A33,Th78; f9.:(the carrier of G9) = f.:(the carrier of G9) & H9 is strict StableSubgroup of H9 by Lm4,RELAT_1:129; then Image f9 = H9 by A2,Def22; then A39: rng f9 = the carrier of H9 by Th49; then reconsider H99 as normal StableSubgroup of H9 by A29,A36,FUNCT_2:def 3; A40: N2 = L9"\/"N9 by A35,A37,Lm5; hence L"\/"(G9/\N) is normal StableSubgroup of G9 by A29,A36,A38,A39,Th86, FUNCT_2:def 3; set l = nat_hom H99; set f1 = l*f99; A41: N2 = L"\/"(G9/\N) by A40,Th86; A42: L"\/"N is StableSubgroup of G9"\/"N by A29,Th38; A43: now let x; assume A44: x in L * N; then consider g1,g2 be Element of G such that A45: x = g1*g2 and A46: g1 in carr L and A47: g2 in carr N; A48: g2 in N by A47,STRUCT_0:def 5; f.x = f.g1*f.g2 by A45,GROUP_6:def 6 .= f.g1*1_H by A1,A48,Th47 .= f.g1 by GROUP_1:def 4; then A49: f.x in f.:(the carrier of L) by A12,A46,FUNCT_1:def 6; L"\/"N is Subgroup of G9"\/"N by A42,Def7; then A50: the carrier of L"\/"N c= the carrier of G9"\/"N by GROUP_2:def 5; A51: x in the carrier of L"\/"N by A30,A44,Th30; then x in G9"\/"N by A50,STRUCT_0:def 5; then x in G by Th1; then x in dom f by A12,STRUCT_0:def 5; then x in f"(f.:(the carrier of L)) by A49,FUNCT_1:def 7; then x in ((the carrier of G9"\/"N) /\ f"(f.:(the carrier of L))) by A51 ,A50,XBOOLE_0:def 4; hence x in f99"(f.:(the carrier of L)) by FUNCT_1:70; end; L is Subgroup of G9 by A29,Def7; then the carrier of L c= the carrier of G9 by GROUP_2:def 5; then f9.:(the carrier of L) = f.:(the carrier of L) by RELAT_1:129; then f99"(f9.:(the carrier of L)) = L * N by A21,A43,TARSKI:1; then A52: f99"(the carrier of H99) = the carrier of N1 by A34,A30,Th30; A53: f99"(the carrier of Ker l) = f99"(the carrier of H99) by Th48; then the carrier of Ker f1 = the carrier of N1 by A52,Th88; hence L"\/"N is normal StableSubgroup of G9"\/"N by Lm5; A54: Ker f1 = N1 by A52,A53,Lm5,Th88; now set f2 = l*f9; let N19 be strict normal StableSubgroup of G9"\/"N; let N29 be strict normal StableSubgroup of G9; assume A55: N19=L"\/"N; f99.:(the carrier of G9"\/"N) = f9.:(the carrier of G9) & f1.:(the carrier of G9"\/"N) = l.:(f99.:(the carrier of G9"\/"N)) by A2,A20,RELAT_1:126 ,129; then A56: f1.:(the carrier of G9"\/"N)=f2.:(the carrier of G9) by RELAT_1:126; A57: f9"(the carrier of Ker l) = f9"(the carrier of H99) by Th48; assume N29=L"\/"(G9/\N); then A58: N29=Ker f2 by A37,A41,A57,Lm5,Th88; the carrier of Image f1=f1.:(the carrier of G9"\/"N) by Def22 .= the carrier of Image f2 by A56,Def22; then A59: Image f1 = Image f2 by Lm5; (G9"\/"N)./.Ker f1, Image f1 are_isomorphic & Image f2, G9./.Ker f2 are_isomorphic by Th59; hence (G9"\/"N)./.N19,G9./.N29 are_isomorphic by A54,A55,A59,A58,Th55; end; hence thesis; end; :: ALG I.4.7 Lemma 1 begin :: The Zassenhaus Butterfly Lemma theorem Th91: for H,K,H9,K9 being strict StableSubgroup of G, JH being normal StableSubgroup of H9"\/"(H/\K), HK being normal StableSubgroup of H/\K st H9 is normal StableSubgroup of H & K9 is normal StableSubgroup of K & JH = H9"\/"(H/\ K9) & HK=(H9/\K)"\/"(K9/\H) holds (H9"\/"(H/\K))./.JH, (H/\K)./.HK are_isomorphic proof let H,K,H9,K9 be strict StableSubgroup of G; reconsider GG = H as GroupWithOperators of O; set G9=H/\K; set L=H/\K9; reconsider G9 as strict StableSubgroup of GG by Lm34; let JH be normal StableSubgroup of H9"\/"(H/\K); let HK be normal StableSubgroup of H/\K; assume that A1: H9 is normal StableSubgroup of H and A2: K9 is normal StableSubgroup of K; A3: L is normal StableSubgroup of G9 by A2,Th60; reconsider N9 = H9 as normal StableSubgroup of GG by A1; assume that A4: JH = H9"\/"(H/\K9) and A5: HK=(H9/\K)"\/"(K9/\H); reconsider N = N9 as StableSubgroup of GG; set N1=G9/\N; A6: G9"\/"N = (H/\K) "\/" H9 by Th86 .= H9"\/"(H/\K); reconsider L as StableSubgroup of GG by A3,Th11; N1=(H/\K)/\H9 by Th39; then A7: L"\/"N1 = (H/\K9)"\/"((H/\K)/\H9) by Th86 .= ((H9/\H)/\K)"\/"(K9/\H) by Th20 .= HK by A1,A5,Lm22; reconsider HH = GG./.N9 as GroupWithOperators of O; reconsider f = nat_hom N9 as Homomorphism of GG,HH; A8: N = Ker f by Th48; L"\/"N = (H/\K9)"\/"H9 by Th86 .=JH by A4; hence thesis by A3,A7,A8,A6,Th90; end; theorem Th92: for H,K,H9,K9 being strict StableSubgroup of G st H9 is normal StableSubgroup of H & K9 is normal StableSubgroup of K holds H9"\/"(H/\K9) is normal StableSubgroup of H9"\/"(H/\K) proof let H,K,H9,K9 be strict StableSubgroup of G; reconsider GG=H as GroupWithOperators of O; reconsider G9=H/\K as strict StableSubgroup of GG by Lm34; assume that A1: H9 is normal StableSubgroup of H and A2: K9 is normal StableSubgroup of K; reconsider N9=H9 as normal StableSubgroup of GG by A1; reconsider N=N9 as StableSubgroup of GG; reconsider HH=GG./.N9 as GroupWithOperators of O; reconsider f=nat_hom N9 as Homomorphism of GG,HH; set L=H/\K9; A3: L is strict normal StableSubgroup of G9 by A2,Th60; then reconsider L as strict StableSubgroup of GG by Th11; A4: N = Ker f by Th48; A5: G9"\/"N = (H/\K)"\/"H9 by Th86 .= H9"\/"(H/\K); L"\/"N = (H/\K9)"\/"H9 by Th86 .= H9"\/"(H/\K9); hence thesis by A3,A4,A5,Th90; end; ::$N Zassenhaus Lemma theorem Th93: for H,K,H9,K9 being strict StableSubgroup of G, JH being normal StableSubgroup of H9"\/"(H/\K), JK being normal StableSubgroup of K9"\/"(K/\H) st JH = H9"\/"(H/\K9) & JK= K9"\/"(K/\H9) & H9 is normal StableSubgroup of H & K9 is normal StableSubgroup of K holds (H9"\/"(H/\K))./.JH, (K9"\/"(K/\H))./.JK are_isomorphic proof let H,K,H9,K9 be strict StableSubgroup of G; let JH be normal StableSubgroup of H9"\/"(H/\K); let JK be normal StableSubgroup of K9"\/"(K/\H); assume that A1: JH = H9"\/"(H/\K9) and A2: JK= K9"\/"(K/\H9); set HK=(H9/\K)"\/"(K9/\H); assume A3: H9 is normal StableSubgroup of H; then A4: H9/\K is normal StableSubgroup of H/\K by Th60; assume A5: K9 is normal StableSubgroup of K; then K9/\H is normal StableSubgroup of H/\K by Th60; then reconsider HK as normal StableSubgroup of H/\K by A4,Th87; HK=(K9/\H)"\/"(H9/\K); then A6: (K9"\/"(K/\H))./.JK, (H/\K)./.HK are_isomorphic by A2,A3,A5,Th91; (H9"\/"(H/\K))./.JH, (H/\K)./.HK are_isomorphic by A1,A3,A5,Th91; hence thesis by A6,Th55; end; begin :: Composition Series :: ALG I.4.7 Definition 9 definition let O be set; let G be GroupWithOperators of O; let IT be FinSequence of the_stable_subgroups_of G; attr IT is composition_series means :Def28: IT.1=(Omega).G & IT.(len IT)= (1).G & for i being Nat st i in dom IT & i+1 in dom IT for H1,H2 being StableSubgroup of G st H1=IT.i & H2=IT.(i+1) holds H2 is normal StableSubgroup of H1; end; registration let O be set; let G be GroupWithOperators of O; cluster composition_series for FinSequence of the_stable_subgroups_of G; existence proof take H=<*(Omega).G,(1).G*>; (Omega).G is Element of the_stable_subgroups_of G & (1).G is Element of the_stable_subgroups_of G by Def11; then reconsider H as FinSequence of the_stable_subgroups_of G by FINSEQ_2:13; A1: H.(len H) = H.2 by FINSEQ_1:44 .=(1).G by FINSEQ_1:44; A2: for i being Nat st i in dom H & i+1 in dom H for H1,H2 being StableSubgroup of G st H1=H.i & H2=H.(i+1) holds H2 is normal StableSubgroup of H1 proof let i be Nat; assume A3: i in dom H; assume A4: i+1 in dom H; len H = 2 by FINSEQ_1:44; then A5: dom H = {1,2} by FINSEQ_1:2,def 3; per cases by A3,A5,TARSKI:def 2; suppose A6: i=1; let H1,H2 be StableSubgroup of G; assume H1=H.i; assume H2=H.(i+1); then A7: H2=(1).G by A6,FINSEQ_1:44; then reconsider H2 as StableSubgroup of H1 by Th16; now let H be strict Subgroup of H1; reconsider H1 as Subgroup of G by Def7; assume the multMagma of H2 = H; then the carrier of H = {1_G} by A7,Def8; then the carrier of H = {1_H1} by GROUP_2:44; then H = (1).H1 by GROUP_2:def 7; hence H is normal; end; hence thesis by Def10; end; suppose i=2; hence thesis by A4,A5,TARSKI:def 2; end; end; H.1=(Omega).G by FINSEQ_1:44; hence thesis by A1,A2,Def28; end; end; definition let O be set; let G be GroupWithOperators of O; mode CompositionSeries of G is composition_series FinSequence of the_stable_subgroups_of G; end; :: ALG I.4.7 Definition 9 definition let O be set; let G be GroupWithOperators of O; let s1,s2 be CompositionSeries of G; pred s1 is_finer_than s2 means :Def29: ex x being set st x c= dom s1 & s2 = s1 * Sgm x; reflexivity proof now let s1 be CompositionSeries of G; set x=dom s1; reconsider x as set; take x; thus x c= dom s1; set i=len s1; Sgm x = Sgm Seg i by FINSEQ_1:def 3 .= idseq i by FINSEQ_3:48; hence s1 = s1 * Sgm x by FINSEQ_2:54; end; hence thesis; end; end; definition let O be set; let G be GroupWithOperators of O; let IT be CompositionSeries of G; attr IT is strictly_decreasing means :Def30: for i being Nat st i in dom IT & i+1 in dom IT for H being StableSubgroup of G, N being normal StableSubgroup of H st H=IT.i & N=IT.(i+1) holds H./.N is not trivial; end; :: ALG I.4.7 Definition 10 definition let O be set; let G be GroupWithOperators of O; let IT be CompositionSeries of G; attr IT is jordan_holder means :Def31: IT is strictly_decreasing & not ex s being CompositionSeries of G st s<>IT & s is strictly_decreasing & s is_finer_than IT; end; :: ALG I.4.7 Definition 9 definition let O be set; let G1,G2 be GroupWithOperators of O; let s1 be CompositionSeries of G1; let s2 be CompositionSeries of G2; pred s1 is_equivalent_with s2 means :Def32: len s1 = len s2 & for n being Nat st n + 1 = len s1 holds ex p being Permutation of Seg n st for H1 being StableSubgroup of G1, H2 being StableSubgroup of G2, N1 being normal StableSubgroup of H1, N2 being normal StableSubgroup of H2, i,j being Nat st 1 <=i & i<=n & j=p.i & H1=s1.i & H2=s2.j & N1=s1.(i+1) & N2=s2.(j+1) holds H1./. N1,H2./.N2 are_isomorphic; end; :: ALG I.4.7 Definition 9 definition let O be set; let G be GroupWithOperators of O; let s be CompositionSeries of G; func the_series_of_quotients_of s -> FinSequence means :Def33: len s = len it + 1 & for i being Nat st i in dom it for H being StableSubgroup of G, N being normal StableSubgroup of H st H=s.i & N=s.(i+1) holds it.i = H./.N if len s > 1 otherwise it={}; existence proof now set i=len s - 1; assume len s > 1; then len s - 1 > 1 - 1 by XREAL_1:9; then reconsider i as Element of NAT by INT_1:3; defpred P[set,set] means for H being StableSubgroup of G, N being normal StableSubgroup of H, j being Nat st $1 in Seg i & j=$1 & H=s.j & N=s.(j+1) holds $2 = H./.N; A1: for k being Nat st k in Seg i ex x being set st P[k,x] proof let k be Nat; reconsider k1=k as Element of NAT by ORDINAL1:def 12; assume A2: k in Seg i; then A3: 1<=k by FINSEQ_1:1; k<=i by A2,FINSEQ_1:1; then A4: k+1<=len s - 1 + 1 by XREAL_1:6; 0+k<=1+k by XREAL_1:6; then k<=len s by A4,XXREAL_0:2; then k1 in Seg len s by A3; then A5: k in dom s by FINSEQ_1:def 3; 1+1<=k+1 by A3,XREAL_1:6; then 1<=k+1 by XXREAL_0:2; then k1+1 in Seg len s by A4; then A6: k+1 in dom s by FINSEQ_1:def 3; then reconsider H=s.k, N=s.(k+1) as Element of the_stable_subgroups_of G by A5, FINSEQ_2:11; reconsider H,N as StableSubgroup of G by Def11; reconsider N as normal StableSubgroup of H by A5,A6,Def28; take H./.N; thus thesis; end; consider f be FinSequence such that A7: dom f = Seg i & for k being Nat st k in Seg i holds P[k,f.k] from FINSEQ_1:sch 1(A1); take f; len f = i by A7,FINSEQ_1:def 3; hence len s = len f + 1; let j be Nat; assume A8: j in dom f; let H be StableSubgroup of G; let N be normal StableSubgroup of H; assume A9: H=s.j; assume N=s.(j+1); hence f.j = H./.N by A7,A8,A9; end; hence thesis; end; uniqueness proof let f1,f2 be FinSequence; now assume len s > 1; assume A10: len s = len f1 + 1; assume A11: for i being Nat st i in dom f1 for H1 being StableSubgroup of G , N1 being normal StableSubgroup of H1 st H1=s.i & N1=s.(i+1) holds f1.i = H1 ./.N1; assume A12: len s = len f2 + 1; assume A13: for i being Nat st i in dom f2 for H1 being StableSubgroup of G , N1 being normal StableSubgroup of H1 st H1=s.i & N1=s.(i+1) holds f2.i = H1 ./.N1; for k being Nat st 1 <=k & k <= len f1 holds f1.k=f2.k proof let k be Nat; reconsider k1 = k as Element of NAT by ORDINAL1:def 12; assume that A14: 1 <=k and A15: k <= len f1; A16: k+1<=len s - 1 + 1 by A10,A15,XREAL_1:6; 0+k<=1+k by XREAL_1:6; then k<=len s by A16,XXREAL_0:2; then k1 in Seg len s by A14; then A17: k in dom s by FINSEQ_1:def 3; 1+1<=k+1 by A14,XREAL_1:6; then 1<=k+1 by XXREAL_0:2; then k1+1 in Seg len s by A16; then A18: k+1 in dom s by FINSEQ_1:def 3; then reconsider H1=s.k,N1=s.(k+1) as Element of the_stable_subgroups_of G by A17,FINSEQ_2:11; reconsider H1,N1 as StableSubgroup of G by Def11; reconsider N1 as normal StableSubgroup of H1 by A17,A18,Def28; A19: k1 in Seg len f1 by A14,A15; then k in dom f1 by FINSEQ_1:def 3; then A20: f1.k=H1./.N1 by A11; k in dom f2 by A10,A12,A19,FINSEQ_1:def 3; hence thesis by A13,A20; end; hence f1=f2 by A10,A12,FINSEQ_1:14; end; hence thesis; end; consistency; end; definition let O be set; let f1,f2 be FinSequence; let p be Permutation of dom f1; pred f1,f2 are_equivalent_under p,O means :Def34: len f1 = len f2 & for H1, H2 being GroupWithOperators of O, i,j being Nat st i in dom f1 & j=p".i & H1=f1 .i & H2=f2.j holds H1,H2 are_isomorphic; end; reserve y for set, H19,H29 for StableSubgroup of G, N19 for normal StableSubgroup of H19, s1,s19,s2,s29 for CompositionSeries of G, fs for FinSequence of the_stable_subgroups_of G, f1,f2 for FinSequence, i,j,n for Nat; theorem Th94: i in dom s1 & i+1 in dom s1 & s1.i=s1.(i+1) & fs=Del(s1,i) implies fs is composition_series proof assume A1: i in dom s1; then consider k be Nat such that A2: len s1 = k + 1 and A3: len Del(s1,i) = k by FINSEQ_3:104; assume i+1 in dom s1; then i+1 in Seg len s1 by FINSEQ_1:def 3; then A4: i+1<=len s1 by FINSEQ_1:1; assume A5: s1.i=s1.(i+1); assume A6: fs = Del(s1,i); A7: i in Seg len s1 by A1,FINSEQ_1:def 3; then A8: 1<=i by FINSEQ_1:1; then 1+1<=i+1 by XREAL_1:6; then 1+1 <= len fs+1 by A6,A4,A2,A3,XXREAL_0:2; then A9: 1 <= len fs by XREAL_1:6; per cases by A9,XXREAL_0:1; suppose A10: len fs = 1; A11: now let n be Nat; assume n in dom fs; then n in Seg 1 by A10,FINSEQ_1:def 3; then A12: n=1 by FINSEQ_1:2,TARSKI:def 1; assume A13: n+1 in dom fs; let H1,H2; assume that H1=fs.n and H2=fs.(n+1); 2 in Seg 1 by A10,A12,A13,FINSEQ_1:def 3; hence H2 is normal StableSubgroup of H1 by FINSEQ_1:2,TARSKI:def 1; end; A14: s1.1=(Omega).G by Def28; A15: 1<=i by A7,FINSEQ_1:1; A16: i<=1 by A6,A4,A2,A3,A10,XREAL_1:6; then A17: i=1 by A15,XXREAL_0:1; dom s1 = Seg 2 by A6,A2,A3,A10,FINSEQ_1:def 3; then 1 in dom s1; then A18: i in dom s1 by A15,A16,XXREAL_0:1; i<=1 by A6,A4,A2,A3,A10,XREAL_1:6; then A19: fs.(len fs) = s1.(1+1) by A6,A2,A3,A10,A18,FINSEQ_3:111 .= (1).G by A6,A2,A3,A10,Def28; s1.2=(1).G by A6,A2,A3,A10,Def28; hence thesis by A5,A10,A17,A14,A19,A11,Def28; end; suppose A20: len fs > 1; A21: fs.1=(Omega).G proof per cases by A8,XXREAL_0:1; suppose A22: i=1; then fs.1 = s1.(1+1) by A1,A6,A2,A3,A20,FINSEQ_3:111; hence thesis by A5,A22,Def28; end; suppose A23: i>1; reconsider i as Element of NAT by INT_1:3; fs.1 = Del(s1,i).1 by A6 .= s1.1 by A23,FINSEQ_3:110; hence thesis by Def28; end; end; A24: now let n be Nat; assume that A25: n in dom fs and A26: n+1 in dom fs; A27: n in Seg len fs by A25,FINSEQ_1:def 3; then A28: n <= k by A6,A3,FINSEQ_1:1; reconsider n1=n+1 as Nat; A29: n+1 in Seg len fs by A26,FINSEQ_1:def 3; then A30: n1 <= k by A6,A3,FINSEQ_1:1; A31: 0+len fs < 1+len fs by XREAL_1:6; then A32: Seg len fs c= Seg len s1 by A6,A2,A3,FINSEQ_1:5; then n in Seg len s1 by A27; then A33: n in dom s1 by FINSEQ_1:def 3; n1 in Seg len s1 by A29,A32; then A34: n1 in dom s1 by FINSEQ_1:def 3; n1 <= len fs by A29,FINSEQ_1:1; then n1 < len s1 by A6,A2,A3,A31,XXREAL_0:2; then n1 + 1 <= k + 1 by A2,NAT_1:13; then Seg(n1+1) c= Seg len s1 by A2,FINSEQ_1:5; then A35: Seg(n1+1) c= dom s1 by FINSEQ_1:def 3; A36: n1+1 in Seg(n1+1) by FINSEQ_1:4; let H1, H2; assume A37: H1 = fs.n; assume A38: H2 = fs.(n+1); reconsider i,n as Nat; per cases; suppose A39: n=i; reconsider n9=n,i as Element of NAT by INT_1:3; A45: Del(s1,i).n9 = s1.(n9+1) by A1,A2,A28,A44,FINSEQ_3:111; reconsider n19=n1,i,k as Element of NAT by INT_1:3; 0+n<=n+1 by XREAL_1:6; then A46: i<=n19 by A44,XXREAL_0:2; n19<=k by A6,A3,A29,FINSEQ_1:1; then Del(s1,i).n19 = s1.(n19+1) by A1,A2,A46,FINSEQ_3:111; hence H2 is normal StableSubgroup of H1 by A6,A37,A38,A34,A35,A36,A45 ,Def28; end; end; i<=len fs by A6,A4,A2,A3,XREAL_1:6; then fs.(len fs) = s1.(len s1) by A1,A6,A2,A3,FINSEQ_3:111; then fs.(len fs)=(1).G by Def28; hence thesis by A21,A24,Def28; end; end; theorem Th95: s1 is_finer_than s2 implies ex n st len s1 = len s2 + n proof set n=len s1 - len s2; assume s1 is_finer_than s2; then consider x such that A1: x c= dom s1 and A2: s2 = s1 * Sgm x by Def29; A3: x c= Seg len s1 by A1,FINSEQ_1:def 3; reconsider x as finite set by A1; now let y1 be set; assume y1 in dom s2; then y1 in dom Sgm x by A2,FUNCT_1:11; then A4: y1 in Seg card x by A3,FINSEQ_3:40; card x <= card dom s1 by A1,NAT_1:43; then Seg card x c= Seg card dom s1 by FINSEQ_1:5; then y1 in Seg card dom s1 by A4; then y1 in Seg card Seg len s1 by FINSEQ_1:def 3; then y1 in Seg len s1 by FINSEQ_1:57; hence y1 in dom s1 by FINSEQ_1:def 3; end; then dom s2 c= dom s1 by TARSKI:def 3; then Seg len s2 c= dom s1 by FINSEQ_1:def 3; then Seg len s2 c= Seg len s1 by FINSEQ_1:def 3; then len s2 <= len s1 by FINSEQ_1:5; then len s2 - len s2 <= len s1 - len s2 by XREAL_1:9; then n in NAT by INT_1:3; then reconsider n as Nat; take n; thus thesis; end; theorem Th96: len s2 = len s1 & s2 is_finer_than s1 implies s1 = s2 proof reconsider X = Seg len s2 as finite set; assume len s2 = len s1; then A1: dom s1 = Seg len s2 by FINSEQ_1:def 3 .= dom s2 by FINSEQ_1:def 3; assume s2 is_finer_than s1; then consider x such that A2: x c= dom s2 and A3: s1 = s2 * Sgm x by Def29; set y = X \ x; A4: x c= Seg len s2 by A2,FINSEQ_1:def 3; then x = rng Sgm x by FINSEQ_1:def 13; then A5: dom(s2 * Sgm x)=dom Sgm x by A2,RELAT_1:27; reconsider x,y as finite set by A2; dom Sgm x = Seg len s2 by A3,A1,A5,FINSEQ_1:def 3; then len Sgm x = len s2 by FINSEQ_1:def 3; then A6: card x = len s2 by A4,FINSEQ_3:39; A7: X = X \/ x by A4,XBOOLE_1:12 .= x \/ y by XBOOLE_1:39; card(x \/ y) = (card x) + (card y) by CARD_2:40,XBOOLE_1:79; then len s2 = (card x) + (card y) by A7,FINSEQ_1:57; then y = {} by A6; then Sgm x = idseq len s2 by A7,FINSEQ_3:48; hence thesis by A3,FINSEQ_2:54; end; theorem Th97: s1 is not empty & s2 is_finer_than s1 implies s2 is not empty proof assume A1: s1 is not empty; assume s2 is_finer_than s1; then ex i st len s2 = len s1 + i by Th95; hence thesis by A1; end; theorem Th98: s1 is_finer_than s2 & s1 is jordan_holder & s2 is jordan_holder implies s1=s2 proof assume A1: s1 is_finer_than s2; assume s1 is jordan_holder; then A2: s1 is strictly_decreasing by Def31; assume s2 is jordan_holder & s1 <> s2; hence contradiction by A1,A2,Def31; end; Lm36: for P,R being Relation holds P = (rng P)|`R iff P~ = (R~)|(dom (P~)) proof let P,R be Relation; hereby assume A1: P = (rng P)|`R; now let x,y be set; hereby assume A2: [x,y] in P~; then [y,x] in P by RELAT_1:def 7; then [y,x] in R by A1,RELAT_1:def 12; then A3: [x,y] in R~ by RELAT_1:def 7; x in dom(P~) by A2,XTUPLE_0:def 12; hence [x,y] in R~|(dom (P~)) by A3,RELAT_1:def 11; end; assume A4: [x,y] in R~|(dom (P~)); then [x,y] in R~ by RELAT_1:def 11; then A5: [y,x] in R by RELAT_1:def 7; x in dom(P~) by A4,RELAT_1:def 11; then x in rng P by RELAT_1:20; then [y,x] in (rng P)|`R by A5,RELAT_1:def 12; hence [x,y] in P~ by A1,RELAT_1:def 7; end; hence P~ = (R~)|(dom (P~)) by RELAT_1:def 2; end; assume A6: P~ = (R~)|(dom (P~)); now let x,y be set; hereby assume [x,y] in P; then A7: [y,x] in P~ by RELAT_1:def 7; then [y,x] in R~ by A6,RELAT_1:def 11; then A8: [x,y] in R by RELAT_1:def 7; y in dom(P~) by A6,A7,RELAT_1:def 11; then y in rng P by RELAT_1:20; hence [x,y] in (rng P)|`R by A8,RELAT_1:def 12; end; assume A9: [x,y] in (rng P)|`R; then [x,y] in R by RELAT_1:def 12; then A10: [y,x] in R~ by RELAT_1:def 7; y in rng P by A9,RELAT_1:def 12; then y in dom(P~) by RELAT_1:20; then [y,x] in R~|(dom (P~)) by A10,RELAT_1:def 11; hence [x,y] in P by A6,RELAT_1:def 7; end; hence thesis by RELAT_1:def 2; end; Lm37: for X being set, P,R being Relation holds P*(R|X) = (X|`P)*R proof let X be set; let P,R be Relation; A1: now let x be set; assume A2: x in (X|`P)*R; then consider y,z be set such that A3: x = [y,z] by RELAT_1:def 1; consider w be set such that A4: [y,w] in X|`P and A5: [w,z] in R by A2,A3,RELAT_1:def 8; w in X by A4,RELAT_1:def 12; then A6: [w,z] in R|X by A5,RELAT_1:def 11; [y,w] in P by A4,RELAT_1:def 12; hence x in P*(R|X) by A3,A6,RELAT_1:def 8; end; now let x be set; assume A7: x in P*(R|X); then consider y,z be set such that A8: x = [y,z] by RELAT_1:def 1; consider w be set such that A9: [y,w] in P and A10: [w,z] in R|X by A7,A8,RELAT_1:def 8; w in X by A10,RELAT_1:def 11; then A11: [y,w] in X|`P by A9,RELAT_1:def 12; [w,z] in R by A10,RELAT_1:def 11; hence x in (X|`P)*R by A8,A11,RELAT_1:def 8; end; hence thesis by A1,TARSKI:1; end; Lm38: for n being Nat, X being set, f being PartFunc of REAL, REAL st X c= Seg n & X c= dom f & f|X is increasing & f.:X c= NAT \ {0} holds Sgm(f.:X) = f * Sgm X proof let n be Nat; let X be set; let f be PartFunc of REAL, REAL; assume A1: X c= Seg n; then A2: rng Sgm X = X by FINSEQ_1:def 13; assume A3: X c= dom f; assume A4: f|X is increasing; assume A5: f.:X c= NAT \ {0}; per cases; suppose A6: X misses dom f; then A7: f.:X = {} by RELAT_1:118; then f.:X c= Seg 0; then Sgm(f.:X) = {} by A7,FINSEQ_1:51; hence thesis by A2,A6,RELAT_1:44; end; suppose A8: X meets dom f; reconsider X9=X as finite set by A1; set fX = f.:X; reconsider f9=f as Function; f9.:X9 is finite; then reconsider fX as finite non empty real-membered set by A8,RELAT_1:118; set k = max fX; reconsider k as Nat by A5; set fs = f * Sgm X; rng Sgm X c= dom f by A1,A3,FINSEQ_1:def 13; then reconsider fs as FinSequence by FINSEQ_1:16; f.:(rng Sgm X) c= NAT \ {0} by A1,A5,FINSEQ_1:def 13; then A9: rng fs c= NAT \ {0} by RELAT_1:127; NAT \ {0} c= NAT by XBOOLE_1:36; then rng fs c= NAT by A9,XBOOLE_1:1; then reconsider fs as FinSequence of NAT by FINSEQ_1:def 4; now let x be set; assume A10: x in f.:X; then reconsider k9=x as Nat by A5; not k9 in {0} by A5,A10,XBOOLE_0:def 5; then k9 <> 0 by TARSKI:def 1; then 0+1 < k9+1 by XREAL_1:6; then A11: 1 <= k9 by NAT_1:13; k9 <= k by A10,XXREAL_2:def 8; hence x in Seg k by A11,FINSEQ_1:1; end; then A12: f.:X c= Seg k by TARSKI:def 3; A13: now A14: dom fs = Seg len fs by FINSEQ_1:def 3; let l,m,k1,k2 be Nat; assume that A15: 1 <= l and A16: l < m and A17: m <= len fs; set k19=(Sgm X).l; l <= len fs by A16,A17,XXREAL_0:2; then A18: l in dom fs by A15,A14,FINSEQ_1:1; then l in dom Sgm X by FUNCT_1:11; then A19: k19 in X by A2,FUNCT_1:3; set k29=(Sgm X).m; 1 <= m by A15,A16,XXREAL_0:2; then A20: m in dom fs by A17,A14,FINSEQ_1:1; then A21: m in dom Sgm X by FUNCT_1:11; then A22: k29 in X by A2,FUNCT_1:3; reconsider k19,k29 as Nat; m in Seg len Sgm X by A21,FINSEQ_1:def 3; then m <= len Sgm X by FINSEQ_1:1; then A23: k19 < k29 by A1,A15,A16,FINSEQ_1:def 13; reconsider k19,k29 as Element of NAT by ORDINAL1:def 12; reconsider k19,k29 as Element of REAL; (Sgm X).l in dom f by A18,FUNCT_1:11; then A24: k19 in X /\ dom f by A19,XBOOLE_0:def 4; assume that A25: k1 = fs.l and A26: k2 = fs.m; A27: k2 = f.((Sgm X).m) by A26,A20,FUNCT_1:12; (Sgm X).m in dom f by A20,FUNCT_1:11; then A28: k29 in X /\ dom f by A22,XBOOLE_0:def 4; k1 = f.((Sgm X).l) by A25,A18,FUNCT_1:12; hence k1 < k2 by A4,A27,A23,A24,A28,RFUNCT_2:20; end; rng fs = f.:X by A2,RELAT_1:127; hence thesis by A12,A13,FINSEQ_1:def 13; end; end; Lm39: y c= Seg(n+1) & i in Seg(n+1) & not i in y implies ex x st Sgm x = Sgm( Seg(n+1)\{i})" * Sgm y & x c= Seg n proof set x1 = {k where k is Element of NAT: k in y & ki}; set x = x1 \/ x2; set f1 = id x1; assume A1: y c= Seg(n+1); then A2: y = rng Sgm y by FINSEQ_1:def 13; assume A3: i in Seg(n+1); then A4: 1<=i by FINSEQ_1:1; A5: i<=n+1 by A3,FINSEQ_1:1; A6: now let z be set; assume A7: z in x; per cases by A7,XBOOLE_0:def 3; suppose z in x1; then A8: ex k be Element of NAT st k = z & k in y & k < i; then reconsider z9=z as Element of NAT; z9 i; reconsider z9=z as integer number by A10; 1i}; now let x be set; assume x in f2; then consider k be Element of NAT such that A15: [k-1,k] = x and k in y9 and k > i; reconsider y=k-1,z=k as set; take y,z; thus x = [y,z] by A15; end; then reconsider f2 as Relation by RELAT_1:def 1; set f = f1 \/ f2; A16: now let x be set; assume x in x2; then consider k be Element of NAT such that A17: k-1 = x & k in y9 & k > i; reconsider y=k as set; [x,y] in f2 by A17; hence x in dom f2 by XTUPLE_0:def 12; end; now let x be set; assume x in dom f2; then consider y be set such that A18: [x,y] in f2 by XTUPLE_0:def 12; consider k be Element of NAT such that A19: [k-1,k] = [x,y] and A20: k in y9 & k > i by A18; k-1 = x by A19,XTUPLE_0:1; hence x in x2 by A20; end; then A21: dom f2 = x2 by A16,TARSKI:1; A22: now let x,y1,y2 be set; assume A23: [x,y1] in f; assume A24: [x,y2] in f; per cases by A23,XBOOLE_0:def 3; suppose A25: [x,y1] in f1; then A26: x in dom f1 by XTUPLE_0:def 12; then f1.x = x by FUNCT_1:17; then A27: y1 = x by A25,A26,FUNCT_1:def 2; per cases by A24,XBOOLE_0:def 3; suppose A28: [x,y2] in f1; then A29: x in dom f1 by XTUPLE_0:def 12; then f1.x = x by FUNCT_1:17; hence y1 = y2 by A27,A28,A29,FUNCT_1:def 2; end; suppose A30: [x,y2] in f2; x in x1 by A26; then consider k9 be Element of NAT such that A31: k9=x and k9 in y and A32: k9i; then k9+1>i by A31; hence y1 = y2 by A32,NAT_1:13; end; end; suppose [x,y1] in f2; then consider k be Element of NAT such that A33: [k-1,k] = [x,y1] and k in y9 and A34: k > i; A35: k-1=x by A33,XTUPLE_0:1; per cases by A24,XBOOLE_0:def 3; suppose [x,y2] in f1; then x in dom f1 by XTUPLE_0:def 12; then x in x1; then consider k9 be Element of NAT such that A36: k9=x and k9 in y and A37: k9i by A34; hence y1 = y2 by A37,NAT_1:13; end; suppose [x,y2] in f2; then consider k9 be Element of NAT such that A38: [k9-1,k9] = [x,y2] and k9 in y9 and k9 > i; k9-1=x by A38,XTUPLE_0:1; hence y1 = y2 by A33,A35,A38,XTUPLE_0:1; end; end; end; A39: now let x,y1,y2 be set; assume [x,y1] in f2; then consider k be Element of NAT such that A40: [k-1,k] = [x,y1] and k in y9 and k > i; A41: k-1 = x by A40,XTUPLE_0:1; assume [x,y2] in f2; then consider k9 be Element of NAT such that A42: [k9-1,k9] = [x,y2] and k9 in y9 and k9 > i; k9-1 = x by A42,XTUPLE_0:1; hence y1 = y2 by A40,A42,A41,XTUPLE_0:1; end; reconsider f as Function by A22,FUNCT_1:def 1; A43: now let x be set; A44: f1 c= f by XBOOLE_1:7; dom f = dom f1 \/ dom f2 by RELAT_1:1; then A45: dom f1 c= dom f by XBOOLE_1:7; assume A46: x in dom f1; then [x,f1.x] in f1 by FUNCT_1:def 2; hence f.x = f1.x by A46,A45,A44,FUNCT_1:def 2; end; reconsider f2 as Function by A39,FUNCT_1:def 1; assume A47: not i in y; A48: now let z be set; set k=z; assume A49: z in y9; then k in Seg(n+1) by A1; then reconsider k as Element of NAT; per cases; suppose k <= i; then k < i by A47,A49,XXREAL_0:1; then z in x1 by A49; then z in rng f1; then z in (rng f1 \/ rng f2) by XBOOLE_0:def 3; hence z in rng f by RELAT_1:12; end; suppose A50: k > i; set x99=k-1; [x99,z] in f2 by A49,A50; then z in rng f2 by XTUPLE_0:def 13; then z in (rng f1 \/ rng f2) by XBOOLE_0:def 3; hence z in rng f by RELAT_1:12; end; end; now let z be set; assume z in rng f; then A51: z in (rng f1 \/ rng f2) by RELAT_1:12; per cases by A51,XBOOLE_0:def 3; suppose z in rng f1; then z in x1; then ex k be Element of NAT st k = z & k in y & k < i; hence z in y9; end; suppose z in rng f2; then consider x99 be set such that A52: [x99,z] in f2 by XTUPLE_0:def 13; ex k be Element of NAT st [k-1,k] = [x99,z] & k in y9 & k > i by A52; hence z in y9 by XTUPLE_0:1; end; end; then A53: rng f = y9 by A48,TARSKI:1; now let a,b be set; hereby assume A54: [a,b] in f; per cases by A54,XBOOLE_0:def 3; suppose A55: [a,b] in f1; reconsider i9=i,n9=n as Element of NAT by ORDINAL1:def 12; A56: a = b by A55,RELAT_1:def 10; a in x1 by A55,RELAT_1:def 10; then consider a9 be Element of NAT such that A57: a9 = a and A58: a9 in y and A59: a9i by A63; A67: a = b9-1 by A64,XTUPLE_0:1; reconsider a9=b9-1 as integer number; i+1<=b9 by A66,NAT_1:13; then A68: i+1-1<=b9-1 by XREAL_1:9; then A69: 1<=a9 by A4,XXREAL_0:2; reconsider a9 as Element of NAT by A68,INT_1:3; b9<=n+1 by A1,A65,FINSEQ_1:1; then A70: b9-1<=n+1-1 by XREAL_1:9; then A71: a9 in Seg n by A69; then a in Seg n by A64,XTUPLE_0:1; then a in Seg len Sgm(Seg(n+1)\{i}) by A3,FINSEQ_3:107; then A72: a in dom Sgm(Seg(n+1)\{i}) by FINSEQ_1:def 3; a9+1 = Sgm(Seg(n9+1)\{i9}).a9 by A3,A70,A68,A71,FINSEQ_3:108; then [a,b] in Sgm(Seg(n+1)\{i}) by A64,A67,A72,FUNCT_1:1; hence [a,b] in (rng f)|`Sgm(Seg(n+1)\{i}) by A53,A64,A65,RELAT_1:def 12; end; end; assume A73: [a,b] in (rng f)|`Sgm(Seg(n+1)\{i}); then A74: [a,b] in Sgm(Seg(n+1)\{i}) by RELAT_1:def 12; then A75: a in dom Sgm(Seg(n+1)\{i}) by XTUPLE_0:def 12; b in rng f by A73,RELAT_1:def 12; then b in Seg(n+1) by A1,A53; then reconsider a9=a,b9=b as Element of NAT by A75; A76: a in Seg len Sgm(Seg(n+1)\{i}) by A75,FINSEQ_1:def 3; then A77: 1<=a9 by FINSEQ_1:1; A78: b in y by A53,A73,RELAT_1:def 12; A79: a in Seg n by A3,A76,FINSEQ_3:107; reconsider i,n as Element of NAT by ORDINAL1:def 12; A80: a9<=n by A79,FINSEQ_1:1; per cases; suppose A81: a9i by A83,NAT_1:13; A86: a = b9-1 by A84; b9 in y9 by A53,A73,RELAT_1:def 12; then [a,b] in f2 by A85,A86; hence [a,b] in f by XBOOLE_0:def 3; end; end; then A87: f = (rng f)|`Sgm(Seg(n+1)\{i}) by RELAT_1:def 2; reconsider g=f" as PartFunc of dom(f"), rng(f") by RELSET_1:4; A88: now let x be set; A89: f2 c= f by XBOOLE_1:7; dom f = dom f1 \/ dom f2 by RELAT_1:1; then A90: dom f2 c= dom f by XBOOLE_1:7; assume A91: x in dom f2; then [x,f2.x] in f2 by FUNCT_1:def 2; hence f.x = f2.x by A91,A90,A89,FUNCT_1:def 2; end; now let y1,y2 be set; assume y1 in dom f; then A92: y1 in dom f1 \/ dom f2 by RELAT_1:1; assume y2 in dom f; then A93: y2 in dom f1 \/ dom f2 by RELAT_1:1; assume A94: f.y1 = f.y2; per cases by A92,XBOOLE_0:def 3; suppose A95: y1 in dom f1; then A96: f1.y1 = y1 by FUNCT_1:17; then A97: f.y1 = y1 by A43,A95; per cases by A93,XBOOLE_0:def 3; suppose A98: y2 in dom f1; then f1.y2 = y2 by FUNCT_1:17; hence y1 = y2 by A43,A94,A97,A98; end; suppose A99: y2 in dom f2; then f.y2 = f2.y2 by A88; then [y2,f.y2] in f2 by A99,FUNCT_1:def 2; then A100: ex k be Element of NAT st [k-1,k] = [y2,f.y2] & k in y9 & k>i; f.y1 = f1.y1 by A43,A95; then f.y1 in x1 by A95,A96; then ex k9 be Element of NAT st k9=f.y1 & k9 in y & k9i; A104: k = f.y1 by A102,XTUPLE_0:1; per cases by A93,XBOOLE_0:def 3; suppose A105: y2 in dom f1; then f1.y2 = y2 by FUNCT_1:17; then f.y2 in dom f1 by A43,A105; then f.y2 in x1; then ex k9 be Element of NAT st k9=f.y2 & k9 in y & k9i; k = f.y2 by A107,XTUPLE_0:1; hence y1 = y2 by A94,A102,A104,A107,XTUPLE_0:1; end; end; end; then A108: f is one-to-one by FUNCT_1:def 4; then f" = f~ by FUNCT_1:def 5; then A109: f" = (Sgm(Seg(n+1)\{i})~)|dom(f") by A87,Lm36; dom f1 = x1; then A110: dom f = x9 by A21,RELAT_1:1; then dom f c= NAT by A14,XBOOLE_1:1; then rng g c= NAT by A108,FUNCT_1:33; then A111: rng g c= REAL by XBOOLE_1:1; rng f c= NAT by A1,A53,XBOOLE_1:1; then dom g c= NAT by A108,FUNCT_1:33; then dom g c= REAL by XBOOLE_1:1; then reconsider g as PartFunc of REAL,REAL by A111,RELSET_1:7; A112: dom(f") = y by A108,A53,FUNCT_1:33; now let r1,r2 be Element of REAL; A113: g = (f1 \/ f2)~ by A108,FUNCT_1:def 5 .= f1~ \/ f2~ by RELAT_1:23; assume r1 in y /\ dom g; then A114: [r1,g.r1] in g by A112,FUNCT_1:1; assume r2 in y /\ dom g; then A115: [r2,g.r2] in g by A112,FUNCT_1:1; assume A116: r1 < r2; per cases by A114,A113,XBOOLE_0:def 3; suppose [r1,g.r1] in f1~; then A117: [r1,g.r1] in id x1; then A118: r1=g.r1 by RELAT_1:def 10; r1 in x1 by A117,RELAT_1:def 10; then A119: ex k9 be Element of NAT st g.r1=k9 & k9 in y & k9i; reconsider k999=g.r2,i9=i-1 as integer number by A120,XTUPLE_0:1; k99-1=g.r2 by A120,XTUPLE_0:1; then i-1 < g.r2 by A121,XREAL_1:9; then i9+1<=k999 by INT_1:7; hence g.r1 < g.r2 by A119,XXREAL_0:2; end; end; suppose [r1,g.r1] in f2~; then [g.r1,r1] in f2 by RELAT_1:def 7; then consider k9 be Element of NAT such that A122: [k9-1,k9]=[g.r1,r1] and k9 in y9 and A123: k9>i; A124: k9-1=g.r1 by A122,XTUPLE_0:1; A125: r1=k9 by A122,XTUPLE_0:1; per cases by A115,A113,XBOOLE_0:def 3; suppose [r2,g.r2] in f1~; then [r2,g.r2] in id x1; then r2 in x1 by RELAT_1:def 10; then ex k99 be Element of NAT st r2=k99 & k99 in y & k99i; k99-1=g.r2 & r2=k99 by A126,XTUPLE_0:1; hence g.r1 < g.r2 by A116,A124,A125,XREAL_1:9; end; end; end; then A127: g|y is increasing by RFUNCT_2:20; A128: rng(f") = x by A108,A110,FUNCT_1:33; then A129: x = (f").: y by A112,RELAT_1:113; now let x9 be set; assume A130: x9 in g.:y; then not x9=0 by A14,A129,FINSEQ_1:1; then A131: not x9 in {0} by TARSKI:def 1; x9 in Seg n by A6,A129,A130; hence x9 in NAT \ {0} by A131,XBOOLE_0:def 5; end; then A132: g.:y c= NAT \ {0} by TARSKI:def 3; take x; Sgm(Seg(n+1)\{i}) is one-to-one by FINSEQ_3:92,XBOOLE_1:36; then A133: Sgm(Seg(n+1)\{i})"=Sgm(Seg(n+1)\{i})~ by FUNCT_1:def 5; Sgm x = Sgm (g.:y) by A112,A128,RELAT_1:113 .= ((Sgm(Seg(n+1)\{i})")|y) * Sgm y by A1,A112,A127,A132,A133,A109,Lm38 .= Sgm(Seg(n+1)\{i})" * y|`Sgm y by Lm37 .= Sgm(Seg(n+1)\{i})" * Sgm y by A2,RELAT_1:95; hence Sgm x = Sgm(Seg(n+1)\{i})" * Sgm y; thus thesis by A6,TARSKI:def 3; end; theorem Th99: i in dom s1 & i+1 in dom s1 & s1.i = s1.(i+1) & s19 = Del(s1,i) & s2 is jordan_holder & s1 is_finer_than s2 implies s19 is_finer_than s2 proof assume that A1: i in dom s1 and A2: i+1 in dom s1; A3: i in Seg len s1 by A1,FINSEQ_1:def 3; then A4: 1 <= i by FINSEQ_1:1; set k = len s1 - 1; assume A5: s1.i = s1.(i+1); reconsider k as integer number; assume A6: s19 = Del(s1,i); assume A7: s2 is jordan_holder; i<=len s1 by A3,FINSEQ_1:1; then 1 <= len s1 by A4,XXREAL_0:2; then 1-1 <= len s1 - 1 by XREAL_1:9; then reconsider k as Element of NAT by INT_1:3; A8: dom s1 = Seg(k+1) by FINSEQ_1:def 3; assume s1 is_finer_than s2; then consider z be set such that A9: z c= dom s1 and A10: s2 = s1 * Sgm z by Def29; A11: i+1 in Seg len s1 by A2,FINSEQ_1:def 3; now per cases; suppose A12: not i in z; set y = z; take y; thus y c= Seg(k+1) by A9,FINSEQ_1:def 3; thus not i in y by A12; thus s2 = s1 * Sgm y by A10; end; suppose A13: i in z; now let x; assume A14: x in {i+1} /\ {i}; then x in {i} by XBOOLE_0:def 4; then A15: x=i by TARSKI:def 1; x in {i+1} by A14,XBOOLE_0:def 4; then x=i+1 by TARSKI:def 1; hence contradiction by A15; end; then {i+1} /\ {i} = {} by XBOOLE_0:def 1; then A16: {i+1} misses {i} by XBOOLE_0:def 7; reconsider y = (z \/ {i+1}) \ {i} as set; take y; {i+1} c= Seg(k+1) by A11,ZFMISC_1:31; then A17: z \/ {i+1} c= Seg(k+1) by A9,A8,XBOOLE_1:8; hence A18: y c= Seg(k+1) by XBOOLE_1:1; then y c= dom s1 by FINSEQ_1:def 3; then A19: rng Sgm y c= dom s1 by A18,FINSEQ_1:def 13; reconsider y9=y,z as finite set by A9; A20: dom Sgm y9 = Seg card y9 by A17,FINSEQ_3:40,XBOOLE_1:1; i in z \/ {i+1} by A13,XBOOLE_0:def 3; then {i} c= z \/ {i+1} by ZFMISC_1:31; then card((z \/ {i+1})\{i}) = card(z \/ {i+1}) - card{i} by CARD_2:44; then A21: card y9 = card(z \/ {i+1}) - 1 by CARD_1:30; A22: now A23: 0+i < 1+i by XREAL_1:6; assume i+1 in z; then i+1 in rng Sgm z by A9,A8,FINSEQ_1:def 13; then consider x99 be set such that A24: x99 in dom Sgm z and A25: i+1 = Sgm(z).x99 by FUNCT_1:def 3; i in rng Sgm z by A9,A8,A13,FINSEQ_1:def 13; then consider x9 be set such that A26: x9 in dom Sgm z and A27: i = Sgm(z).x9 by FUNCT_1:def 3; reconsider x9,x99 as Element of NAT by A26,A24; A28: dom Sgm z = Seg len Sgm z by FINSEQ_1:def 3; then A29: x9 <= len Sgm z by A26,FINSEQ_1:1; 1 <= x99 by A24,A28,FINSEQ_1:1; then x9 < x99 by A9,A8,A27,A25,A23,A29,FINSEQ_3:41; then reconsider l = x99 - x9 as Element of NAT by INT_1:5; per cases; suppose l = 0; hence contradiction by A27,A25,A23; end; suppose A30: 0 < l; set x999 = x9+1; 0+1 < l+1 by A30,XREAL_1:6; then x9+1-x9 <= x99-x9 by NAT_1:13; then A31: x999 <= x99 by XREAL_1:9; x99 <= len Sgm z by A24,A28,FINSEQ_1:1; then A32: x999 <= len Sgm z by A31,XXREAL_0:2; A33: 1+x9 > 0+x9 & 1 <= x9 by A26,A28,FINSEQ_1:1,XREAL_1:6; then 1 <= x999 by XXREAL_0:2; then x999 in dom Sgm z by A28,A32; then reconsider k3= Sgm(z).x999 as Element of NAT by FINSEQ_2:11; i < k3 by A9,A8,A27,A33,A32,FINSEQ_1:def 13; then A34: i+1 <= k3 by NAT_1:13; A35: 1 <= x999 & x999 < x99 & x99 <= len Sgm z or x999 = x99 by A24,A28 ,A31,A33,FINSEQ_1:1,XXREAL_0:1,2; then A36: x9+1 in dom s2 by A2,A9,A10,A8,A24,A25,A34,FINSEQ_1:def 13,FUNCT_1:11 ; A37: s2 is strictly_decreasing by A7,Def31; A38: x9 in dom s2 by A1,A10,A26,A27,FUNCT_1:11; then reconsider H1=s2.x9,H2=s2.(x9+1) as Element of the_stable_subgroups_of G by A36,FINSEQ_2:11; reconsider H1,H2 as StableSubgroup of G by Def11; reconsider H1 as GroupWithOperators of O; reconsider H2 as normal StableSubgroup of H1 by A38,A36,Def28; s2.x9 = s1.(Sgm(z).x9) by A10,A26,FUNCT_1:13 .= s2.(x9+1) by A5,A9,A10,A8,A27,A24,A25,A35,A34,FINSEQ_1:def 13 ,FUNCT_1:13; then the carrier of H1 = the carrier of H2; then H1./.H2 is trivial by Th77; hence contradiction by A38,A36,A37,Def30; end; end; then card(z \/ {i+1}) = card z + 1 by CARD_2:41; then A39: dom Sgm y9 = dom Sgm z by A9,A8,A21,A20,FINSEQ_3:40; set z2 = {x where x is Element of NAT: x in z & i+1 < x}; set z1 = {x where x is Element of NAT: x in z & x < i}; A40: now let x; assume x in z1 \/ {i} \/ z2; then x in z1 \/ {i} or x in z2 by XBOOLE_0:def 3; then x in z1 or x in {i} or x in z2 by XBOOLE_0:def 3; then consider x9,x99 be Element of NAT such that A41: x = x9 & x9 in z & x9 ^ Sgm z2 by A4,A70,FINSEQ_3:44; z1 \/ {i+1} c= y by A55,XBOOLE_1:7; then z1 \/ {i+1} c= Seg(k+1) by A18,XBOOLE_1:1; then A73: Sgm y = Sgm(z1 \/ {i+1}) ^ Sgm z2 by A55,A57,A63,FINSEQ_3:42; then A74: Sgm y = Sgm z1 ^ <*i+1*> ^ Sgm z2 by A69,FINSEQ_3:44; A75: now let x; A76: len(Sgm z1 ^ <*i*>) = len Sgm z1 + len <*i*> by FINSEQ_1:22 .= len Sgm z1 + 1 by FINSEQ_1:40 .= len Sgm z1 + len <*i+1*> by FINSEQ_1:40 .= len(Sgm z1 ^ <*i+1*>) by FINSEQ_1:22; assume A77: x in dom Sgm z; then reconsider x9=x as Element of NAT; A78: dom(Sgm z1 ^ <*i*>) = Seg len(Sgm z1 ^ <*i*>) by FINSEQ_1:def 3 .= dom(Sgm z1 ^ <*i+1*>) by A76,FINSEQ_1:def 3; per cases by A72,A77,FINSEQ_1:25; suppose A79: x9 in dom(Sgm z1 ^ <*i*>); per cases by A79,FINSEQ_1:25; suppose A80: x9 in dom Sgm z1; then A81: (Sgm z1).x9 = (Sgm z1 ^ <*i*>).x9 by FINSEQ_1:def 7 .= (Sgm z1 ^ <*i*> ^ Sgm z2).x9 by A79,FINSEQ_1:def 7 .= (Sgm z).x9 by A4,A70,A71,FINSEQ_3:44; (Sgm z1).x9 = (Sgm z1 ^ <*i+1*>).x9 by A80,FINSEQ_1:def 7 .= (Sgm z1 ^ <*i+1*> ^ Sgm z2).x9 by A78,A79,FINSEQ_1:def 7 .= (Sgm y).x9 by A73,A69,FINSEQ_3:44; hence (Sgm z).x <> i implies (Sgm y).x = (Sgm z).x by A81; thus (Sgm z).x = i implies (Sgm y).x = i+1 proof assume (Sgm z).x = i; then i in rng Sgm z1 by A80,A81,FUNCT_1:3; then i in z1 by A47,FINSEQ_1:def 13; then ex x999 be Element of NAT st x999=i & x999 in z & x999< i; hence thesis; end; end; suppose ex x99 being Nat st x99 in dom <*i*> & x9=len Sgm z1 + x99; then consider x99 be Nat such that A82: x99 in dom <*i*> and A83: x9=len Sgm z1 + x99; A84: x99 in Seg 1 by A82,FINSEQ_1:38; then A85: x99 = 1 by FINSEQ_1:2,TARSKI:def 1; then i = <*i*>.x99 by FINSEQ_1:40 .= (Sgm z1 ^ <*i*>).x9 by A82,A83,FINSEQ_1:def 7 .= (Sgm z1 ^ <*i*> ^ Sgm z2).x9 by A79,FINSEQ_1:def 7 .= (Sgm z).x9 by A4,A70,A71,FINSEQ_3:44; hence (Sgm z).x <> i implies (Sgm y).x = (Sgm z).x; thus (Sgm z).x = i implies (Sgm y).x = i+1 proof assume (Sgm z).x = i; A86: x99 in dom <*i+1*> by A84,FINSEQ_1:38; i+1 = <*i+1*>.x99 by A85,FINSEQ_1:40 .= (Sgm z1 ^ <*i+1*>).x9 by A83,A86,FINSEQ_1:def 7 .= (Sgm z1 ^ <*i+1*> ^ Sgm z2).x9 by A78,A79,FINSEQ_1:def 7; hence thesis by A73,A69,FINSEQ_3:44; end; end; end; suppose ex x99 being Nat st x99 in dom Sgm z2 & x9=len(Sgm z1 ^ <* i*>) + x99; then consider x99 be Nat such that A87: x99 in dom Sgm z2 and A88: x9=len(Sgm z1 ^ <*i*>) + x99; (Sgm y).x9 = (Sgm z2).x99 by A74,A76,A87,A88,FINSEQ_1:def 7; hence (Sgm z).x <> i implies (Sgm y).x = (Sgm z).x by A72,A87,A88, FINSEQ_1:def 7; thus (Sgm z).x = i implies (Sgm y).x = i+1 proof assume (Sgm z).x = i; then (Sgm z2).x99 = i by A72,A87,A88,FINSEQ_1:def 7; then i in rng Sgm z2 by A87,FUNCT_1:3; then i in z2 by A57,FINSEQ_1:def 13; then ex x999 be Element of NAT st x999=i & x999 in z & i+1< x999; then i+1-i i; (s1 * Sgm y).x = s1.((Sgm y).x) by A91,FUNCT_1:12 .= s1.((Sgm z).x) by A75,A92,A95 .= (s1 * Sgm z).x by A93,FUNCT_1:12; hence (s1 * Sgm y).x = (s1 * Sgm z).x; end; end; hence s2 = s1 * Sgm y by A10,A39,A19,A89,FUNCT_1:2,RELAT_1:27; end; end; then consider y be set such that A96: y c= Seg(k+1) and A97: not i in y and A98: s2 = s1 * Sgm y; now consider x such that A99: Sgm x = Sgm(Seg(k+1)\{i})" * Sgm y and A100: x c= Seg k by A3,A96,A97,Lm39; take x; ex m be Nat st len s1 = m + 1 & len Del(s1,i) = m by A1,FINSEQ_3:104; hence x c= dom s19 by A6,A100,FINSEQ_1:def 3; set f = Sgm (Seg(k+1)\{i}); set X = dom f; set Y = rng f; reconsider f as Function of X,Y by FUNCT_2:1; A101: f is one-to-one by FINSEQ_3:92,XBOOLE_1:36; Seg(k+1)\{i} c= Seg(k+1) by XBOOLE_1:36; then A102: rng f = Seg(k+1)\{i} by FINSEQ_1:def 13; now let x9 be set; assume A103: x9 in y; then not x9 in {i} by A97,TARSKI:def 1; hence x9 in rng f by A96,A102,A103,XBOOLE_0:def 5; end; then y c= rng f by TARSKI:def 3; then A104: rng Sgm y c= rng f by A96,FINSEQ_1:def 13; A105: now 1<=i by A3,FINSEQ_1:1; then A106: 1+1<=i+1 by XREAL_1:6; i+1<=len s1 by A11,FINSEQ_1:1; then 2<=len s1 by A106,XXREAL_0:2; then Seg 2 c= Seg(k+1) by FINSEQ_1:5; then A107: Seg 2\{i} c= rng f by A102,XBOOLE_1:33; assume A108: rng f = {}; per cases by A108,A107,XBOOLE_1:3,ZFMISC_1:58; suppose Seg 2 = {}; hence contradiction; end; suppose Seg 2 = {i}; hence contradiction by FINSEQ_1:2,ZFMISC_1:5; end; end; s19 * Sgm x = s1 * f * Sgm x by A6,FINSEQ_1:def 3 .= s1 * f * f" * Sgm y by A99,RELAT_1:36 .= s1 * (f * f") * Sgm y by RELAT_1:36 .= s1 * id rng f * Sgm y by A101,A105,FUNCT_2:29 .= s1 * (id rng f * Sgm y) by RELAT_1:36 .= s1 * Sgm y by A104,RELAT_1:53; hence s2 = s19 * Sgm x by A98; end; hence thesis by Def29; end; theorem Th100: len s1 > 1 & s2<>s1 & s2 is strictly_decreasing & s2 is_finer_than s1 implies ex i,j st i in dom s1 & i in dom s2 & i+1 in dom s1 & i+1 in dom s2 & j in dom s2 & i+1s2.(i+1) & s1.(i+1) =s2.j proof assume len s1 > 1; then len s1 >= 1+1 by NAT_1:13; then Seg 2 c= Seg len s1 by FINSEQ_1:5; then A1: Seg 2 c= dom s1 by FINSEQ_1:def 3; assume A2: s2<>s1; assume A3: s2 is strictly_decreasing; assume A4: s2 is_finer_than s1; then consider n such that A5: len s2 = len s1 + n by Th95; n<>0 by A2,A4,A5,Th96; then A6: 0 + len s1 < n + len s1 by XREAL_1:6; then Seg len s1 c= Seg len s2 by A5,FINSEQ_1:5; then Seg len s1 c= dom s2 by FINSEQ_1:def 3; then A7: dom s1 c= dom s2 by FINSEQ_1:def 3; now set fX = {k where k is Element of NAT: k in dom s1 & s1.k=s2.k}; A8: 1 in Seg 2; s1.1 = (Omega).G & s2.1 = (Omega).G by Def28; then A9: 1 in fX by A1,A8; now let x; assume x in fX; then ex k be Element of NAT st x=k & k in dom s1 & s1.k=s2.k; hence x in dom s1; end; then fX c= dom s1 by TARSKI:def 3; then reconsider fX as finite non empty real-membered set by A9; set i = max fX; i in fX by XXREAL_2:def 8; then A10: ex k be Element of NAT st i=k & k in dom s1 & s1.k=s2.k; then reconsider i as Element of NAT; take i; thus i in dom s1 & s1.i=s2.i by A10; A11: now assume not i+1 in dom s1; then A12: not i+1 in Seg len s1 by FINSEQ_1:def 3; per cases by A12; suppose 1>i+1; then 1-1>i+1-1 by XREAL_1:9; then 0>i; hence contradiction; end; suppose A13: i+1>len s1; i in Seg len s1 by A10,FINSEQ_1:def 3; then A14: i<=len s1 by FINSEQ_1:1; i>=len s1 by A13,NAT_1:13; then A15: i=len s1 by A14,XXREAL_0:1; then 0+1<=i+1 & i+1<=len s2 by A5,A6,NAT_1:13; then i+1 in Seg len s2; then A16: i+1 in dom s2 by FINSEQ_1:def 3; then reconsider H1=s2.i,H2=s2.(i+1) as Element of the_stable_subgroups_of G by A10,FINSEQ_2:11; reconsider H1,H2 as StableSubgroup of G by Def11; A17: s2.i=(1).G by A10,A15,Def28; then A18: the carrier of H1 = {1_G} by Def8; reconsider H2 as normal StableSubgroup of H1 by A7,A10,A16,Def28; 1_G in H2 by Lm18; then 1_G in the carrier of H2 by STRUCT_0:def 5; then A19: {1_G} c= the carrier of H2 by ZFMISC_1:31; H2 is Subgroup of (1).G by A17,Def7; then the carrier of H2 c= the carrier of (1).G by GROUP_2:def 5; then the carrier of H2 c= {1_G} by Def8; then the carrier of H2 = {1_G} by A19,XBOOLE_0:def 10; then H1./.H2 is trivial by A18,Th77; hence contradiction by A3,A7,A10,A16,Def30; end; end; hence i+1 in dom s1; now A20: 1+i>0+i by XREAL_1:6; assume s1.(i+1)=s2.(i+1); then consider k be Element of NAT such that A21: k>i and A22: k in dom s1 & s1.k=s2.k by A11,A20; k in fX by A22; hence contradiction by A21,XXREAL_2:def 8; end; hence s1.(i+1)<>s2.(i+1); end; then consider i such that A23: i in dom s1 and A24: i+1 in dom s1 and A25: s1.i=s2.i and A26: s1.(i+1)<>s2.(i+1); now consider x such that A27: x c= dom s2 and A28: s1 = s2 * Sgm x by A4,Def29; set j = (Sgm x).(i+1); A29: x c= Seg len s2 by A27,FINSEQ_1:def 3; A30: i+1 in dom Sgm x by A24,A28,FUNCT_1:11; then j in rng Sgm x by FUNCT_1:3; then j in x by A29,FINSEQ_1:def 13; then A31: j in Seg len s2 by A29; then reconsider j as Element of NAT; A32: i+1 <= j by A29,A30,FINSEQ_3:152; take j; thus j in dom s2 by A31,FINSEQ_1:def 3; thus s1.(i+1)=s2.j by A24,A28,FUNCT_1:12; j<>i+1 by A24,A26,A28,FUNCT_1:12; hence i+1s2.(i+1) by A25,A26; thus thesis by A34; end; theorem Th101: i in dom s1 & j in dom s1 & i<=j & H1 = s1.i & H2 = s1.j implies H2 is StableSubgroup of H1 proof assume that A1: i in dom s1 and A2: j in dom s1; defpred P[Nat] means for n,H2 st i+$1 in dom s1 & H2 = s1.(i+$1) holds H2 is StableSubgroup of H1; assume A3: i<=j; assume that A4: H1 = s1.i and A5: H2 = s1.j; A6: for n st P[n] holds P[n+1] proof let n; assume A7: P[n]; set H2 = s1.(i+n); per cases; suppose A8: i+n in dom s1; then reconsider H2 as Element of the_stable_subgroups_of G by FINSEQ_2:11 ; reconsider H2 as StableSubgroup of G by Def11; A9: H2 is StableSubgroup of H1 by A7,A8; now let k be Element of NAT; let H3; assume i+(n+1) in dom s1; then A10: i+n+1 in dom s1; assume H3 = s1.(i+(n+1)); then H3 is StableSubgroup of H2 by A8,A10,Def28; hence H3 is StableSubgroup of H1 by A9,Th11; end; hence thesis; end; suppose not i+n in dom s1; then A11: not i+n in Seg len s1 by FINSEQ_1:def 3; per cases by A11,FINSEQ_1:1; suppose i+n<0+1; then n=0 by NAT_1:13; hence thesis by A1,A4,Def28; end; suppose A12: i+n>len s1; A13: 1+len s1>0+len s1 by XREAL_1:6; i+n+1>len s1+1 by A12,XREAL_1:6; then i+n+1>len s1 by A13,XXREAL_0:2; then not i+n+1 in Seg len s1 by FINSEQ_1:1; hence thesis by FINSEQ_1:def 3; end; end; end; A14: P[0] by A4,Th10; A15: for n holds P[n] from NAT_1:sch 2(A14,A6); set n=j-i; i-i<=j-i by A3,XREAL_1:9; then reconsider n as Element of NAT by INT_1:3; reconsider n as Nat; j=i+n; hence thesis by A2,A5,A15; end; theorem Th102: y in rng the_series_of_quotients_of s1 implies y is strict GroupWithOperators of O proof assume A1: y in rng the_series_of_quotients_of s1; set f1=the_series_of_quotients_of s1; A2: len f1 = 0 or len f1 >= 0+1 by NAT_1:13; per cases by A2; suppose len f1 = 0; then f1 = {}; hence thesis by A1; end; suppose len f1 >= 1; then A3: len s1 > 1 by Def33,CARD_1:27; then A4: len s1 = len f1 + 1 by Def33; consider i be set such that A5: i in dom f1 and A6: f1.i=y by A1,FUNCT_1:def 3; reconsider i as Nat by A5; A7: i in Seg len f1 by A5,FINSEQ_1:def 3; then A8: 1<=i by FINSEQ_1:1; 1<=i by A7,FINSEQ_1:1; then 1+1<=i+1 by XREAL_1:6; then A9: 1<=i+1 by XXREAL_0:2; A10: i<=len f1 by A7,FINSEQ_1:1; then 0+i<=1+i & i+1<=len f1 + 1 by XREAL_1:6; then i<=len s1 by A4,XXREAL_0:2; then i in Seg len s1 by A8,FINSEQ_1:1; then A11: i in dom s1 by FINSEQ_1:def 3; then s1.i in the_stable_subgroups_of G by FINSEQ_2:11; then reconsider H1=s1.i as strict StableSubgroup of G by Def11; i + 1<=len f1 + 1 by A10,XREAL_1:6; then i+1<=len s1 by A3,Def33; then i+1 in Seg len s1 by A9; then A12: i+1 in dom s1 by FINSEQ_1:def 3; then s1.(i+1) in the_stable_subgroups_of G by FINSEQ_2:11; then reconsider N1=s1.(i+1) as strict StableSubgroup of G by Def11; reconsider N1 as normal StableSubgroup of H1 by A11,A12,Def28; y = H1./.N1 by A3,A5,A6,Def33; hence thesis; end; end; theorem Th103: i in dom the_series_of_quotients_of s1 & (for H st H=( the_series_of_quotients_of s1).i holds H is trivial) implies i in dom s1 & i+1 in dom s1 & s1.i=s1.(i+1) proof assume A1: i in dom the_series_of_quotients_of s1; set f1 = the_series_of_quotients_of s1; assume A2: for H st H=(the_series_of_quotients_of s1).i holds H is trivial; A3: len f1 = 0 or len f1 >= 0+1 by NAT_1:13; per cases by A3,XXREAL_0:1; suppose len f1 = 0; then f1 = {}; hence thesis by A1; end; suppose A4: len f1 = 1; f1.i in rng f1 by A1,FUNCT_1:3; then reconsider H=f1.i as strict GroupWithOperators of O by Th102; set H1=(Omega).G; A5: H is trivial by A2; A6: len s1 > 1 by A4,Def33,CARD_1:27; then A7: len s1 = len f1 + 1 by Def33; then A8: s1.2=(1).G by A4,Def28; i in Seg 1 by A1,A4,FINSEQ_1:def 3; then A9: i=1 by FINSEQ_1:2,TARSKI:def 1; then i in Seg 2; hence i in dom s1 by A4,A7,FINSEQ_1:def 3; reconsider N1=(1).G as StableSubgroup of H1 by Th16; A10: s1.1=(Omega).G by Def28; A11: (1).G = (1).H1 by Th15; then reconsider N1 as normal StableSubgroup of H1; A12: H1,H1./.N1 are_isomorphic by A11,Th56; i+1 in Seg 2 by A9; hence i+1 in dom s1 by A4,A7,FINSEQ_1:def 3; for H1, N1 st H1=s1.i & N1=s1.(i+1) holds f1.i = H1./.N1 by A1,A6,Def33; then H1./.N1 is trivial by A10,A8,A9,A5; hence thesis by A10,A8,A9,A11,A12,Th42,Th58; end; suppose A13: len f1 > 1; f1.i in rng f1 by A1,FUNCT_1:3; then reconsider H = f1.i as strict GroupWithOperators of O by Th102; A14: i in Seg len f1 by A1,FINSEQ_1:def 3; then A15: 1<=i by FINSEQ_1:1; 1<=i by A14,FINSEQ_1:1; then 1+1<=i+1 by XREAL_1:6; then A16: 1<=i+1 by XXREAL_0:2; A17: i<=len f1 by A14,FINSEQ_1:1; then A18: 0+i<=1+i & i+1<=len f1 + 1 by XREAL_1:6; A19: len s1 > 1 by A13,Def33,CARD_1:27; then len s1 = len f1 + 1 by Def33; then i<=len s1 by A18,XXREAL_0:2; then A20: i in Seg len s1 by A15,FINSEQ_1:1; hence i in dom s1 by FINSEQ_1:def 3; i + 1<=len f1 + 1 by A17,XREAL_1:6; then i+1<=len s1 by A19,Def33; then A21: i+1 in Seg len s1 by A16; hence i+1 in dom s1 by FINSEQ_1:def 3; A22: i+1 in dom s1 by A21,FINSEQ_1:def 3; then s1.(i+1) in the_stable_subgroups_of G by FINSEQ_2:11; then reconsider N1=s1.(i+1) as strict StableSubgroup of G by Def11; A23: i in dom s1 by A20,FINSEQ_1:def 3; then s1.i in the_stable_subgroups_of G by FINSEQ_2:11; then reconsider H1=s1.i as strict StableSubgroup of G by Def11; reconsider N1 as normal StableSubgroup of H1 by A23,A22,Def28; H is trivial by A2; then H1./.N1 is trivial by A1,A19,Def33; hence thesis by Th76; end; end; theorem Th104: i in dom s1 & i+1 in dom s1 & s1.i=s1.(i+1) & s2=Del(s1,i) implies the_series_of_quotients_of s2=Del(the_series_of_quotients_of s1,i) proof set f1 = the_series_of_quotients_of s1; assume A1: i in dom s1; then consider k be Nat such that A2: len s1 = k + 1 and A3: len Del(s1,i) = k by FINSEQ_3:104; assume i+1 in dom s1; then i+1 in Seg len s1 by FINSEQ_1:def 3; then A4: i+1<=len s1 by FINSEQ_1:1; assume A5: s1.i=s1.(i+1); A6: i in Seg len s1 by A1,FINSEQ_1:def 3; then 1<=i by FINSEQ_1:1; then A7: 1+1<=i+1 by XREAL_1:6; then 2 <= len s1 by A4,XXREAL_0:2; then A8: 1 < len s1 by XXREAL_0:2; then A9: len s1 = len f1 + 1 by Def33; assume A10: s2=Del(s1,i); then 1+1 <= len s2+1 by A7,A4,A2,A3,XXREAL_0:2; then A11: 1 <= len s2 by XREAL_1:6; per cases by A11,XXREAL_0:1; suppose A12: len s2 = 1; then 1 in Seg len f1 by A10,A2,A3,A9; then 1 in dom f1 by FINSEQ_1:def 3; then A13: ex k1 be Nat st len f1 = k1 + 1 & len Del(f1,1) = k1 by FINSEQ_3:104; A14: 1<=i by A6,FINSEQ_1:1; A15: the_series_of_quotients_of s2 = {} by A12,Def33; i<=1 by A10,A4,A2,A3,A12,XREAL_1:6; then len Del(f1,i) = 0 by A10,A2,A3,A9,A12,A13,A14,XXREAL_0:1; hence thesis by A15; end; suppose A16: len s2 > 1; i+1-1<=len s1-1 & 1 <= i by A6,A4,FINSEQ_1:1,XREAL_1:9; then i in Seg len f1 by A9,FINSEQ_1:1; then A17: i in dom f1 by FINSEQ_1:def 3; then consider k1 be Nat such that A18: len f1 = k1 + 1 and A19: len Del(f1,i) = k1 by FINSEQ_3:104; now let n; set n1 = n+1; assume n in dom Del(f1,i); then A20: n in Seg len Del(f1,i) by FINSEQ_1:def 3; then A21: n<=k1 by A19,FINSEQ_1:1; then A22: n1<=k by A2,A9,A18,XREAL_1:6; 1<=n by A20,FINSEQ_1:1; then 1+1<=n+1 by XREAL_1:6; then 1<=n1 by XXREAL_0:2; then n1 in Seg len f1 by A2,A9,A22; then A23: n1 in dom f1 by FINSEQ_1:def 3; reconsider n1 as Nat; let H1, N1; assume A24: H1 = s2.n; 0+n<1+n by XREAL_1:6; then A25: n<=k by A22,XXREAL_0:2; len f1-len Del(f1,i)+len Del(f1,i)>0+len Del(f1,i) by A18,A19,XREAL_1:6; then Seg len Del(f1,i) c= Seg len f1 by FINSEQ_1:5; then n in Seg len f1 by A20; then A26: n in dom f1 by FINSEQ_1:def 3; assume A27: N1 = s2.(n+1); per cases; suppose A28: n=i; reconsider n19=n1 as Element of NAT; 0+i<1+i & n+1>=i+1 by A32,XREAL_1:6; then n1>=i by XXREAL_0:2; then A33: s1.(n19+1) = N1 by A1,A10,A2,A27,A22,FINSEQ_3:111; s1.n19 = H1 by A1,A10,A2,A24,A25,A32,FINSEQ_3:111; then f1.n1 = H1./.N1 by A8,A23,A33,Def33; hence Del(f1,i).n = H1./.N1 by A17,A18,A21,A32,FINSEQ_3:111; end; end; hence thesis by A10,A2,A3,A9,A16,A18,A19,Def33; end; end; theorem f1=the_series_of_quotients_of s1 & i in dom f1 & (for H st H = f1.i holds H is trivial) implies Del(s1,i) is CompositionSeries of G & for s2 st s2 = Del(s1,i) holds the_series_of_quotients_of s2 = Del(f1,i) proof assume A1: f1=the_series_of_quotients_of s1; assume A2: i in dom f1; assume A3: for H st H = f1.i holds H is trivial; then A4: s1.i=s1.(i+1) by A1,A2,Th103; A5: i in dom s1 & i+1 in dom s1 by A1,A2,A3,Th103; hence Del(s1,i) is CompositionSeries of G by A4,Th94,FINSEQ_3:105; let s2; assume s2 = Del(s1,i); hence thesis by A1,A5,A4,Th104; end; theorem Th106: i in dom f1 & (ex p being Permutation of dom f1 st f1,f2 are_equivalent_under p,O & j = p".i) implies ex p9 being Permutation of dom Del(f1,i) st Del(f1,i),Del(f2,j) are_equivalent_under p9,O proof A1: len f1=0 or len f1>=0+1 by NAT_1:13; assume A2: i in dom f1; given p be Permutation of dom f1 such that A3: f1,f2 are_equivalent_under p,O and A4: j = p".i; A5: len f1 = len f2 by A3,Def34; rng(p") c= dom f1; then A6: rng(p") c= Seg len f1 by FINSEQ_1:def 3; p".i in rng(p") by A2,FUNCT_2:4; then p".i in Seg len f1 by A6; then A7: j in dom f2 by A4,A5,FINSEQ_1:def 3; then A8: ex k2 be Nat st len f2 = k2 + 1 & len Del(f2,j) = k2 by FINSEQ_3:104; consider k1 be Nat such that A9: len f1 = k1 + 1 and A10: len Del(f1,i) = k1 by A2,FINSEQ_3:104; per cases by A1,XXREAL_0:1; suppose A11: len f1 = 0; set p9 = the Permutation of dom Del(f1,i); take p9; thus thesis by A9,A11; end; suppose A12: len f1 = 1; reconsider p9={} as Function of dom {}, rng {} by FUNCT_2:1; reconsider p9 as Function of {},{}; A13: p9 is onto by FUNCT_2:def 3; Del(f1,i)={} by A9,A10,A12; then reconsider p9 as Permutation of dom Del(f1,i) by A13; take p9; for H1,H2 being GroupWithOperators of O,l being Nat,n st l in dom Del (f1,i) & n=p9".l & H1=Del(f1,i).l & H2=Del(f2,j).n holds H1,H2 are_isomorphic; hence thesis by A5,A9,A10,A8,Def34; end; suppose A14: len f1 > 1; set Y = (dom f2)\{j}; A15: now assume Y={}; then A16: dom f2 c= {j} by XBOOLE_1:37; {j} c= dom f2 by A7,ZFMISC_1:31; then A17: dom f2 = {j} by A16,XBOOLE_0:def 10; consider k be Nat such that A18: dom f2 = Seg k by FINSEQ_1:def 2; k in NAT by ORDINAL1:def 12; then k = len f2 by A18,FINSEQ_1:def 3; then k >= 1+1 by A5,A14,NAT_1:13; then Seg 2 c= Seg k by FINSEQ_1:5; then { 1,2 } = {j} by A17,A18,FINSEQ_1:2,ZFMISC_1:21; hence contradiction by ZFMISC_1:5; end; set X = (dom f1)\{i}; set p9=(Sgm X)" * p * Sgm Y; Y c= dom f2 by XBOOLE_1:36; then A19: Y c= Seg len f2 by FINSEQ_1:def 3; X c= dom f1 by XBOOLE_1:36; then A20: X c= Seg len f1 by FINSEQ_1:def 3; then A21: rng Sgm X = X by FINSEQ_1:def 13; Y c= dom f2 by XBOOLE_1:36; then Y c= Seg len f2 by FINSEQ_1:def 3; then A22: Sgm Y is one-to-one & rng Sgm Y = Y by FINSEQ_1:def 13,FINSEQ_3:92; A23: dom f1 = Seg len f1 by FINSEQ_1:def 3 .= Seg len f2 by A3,Def34 .= dom f2 by FINSEQ_1:def 3; A24: p.j = (p*p").i by A2,A4,FUNCT_2:15 .= (id dom f1).i by FUNCT_2:61 .= i by A2,FUNCT_1:18; A25: p9 is Permutation of dom Del(f1,i) & p9" = (Sgm Y)" * (p") * (Sgm X) proof set R6=p; set R5=p"; set R4=Sgm X; set R3=(Sgm X)"; set R2=Sgm Y; set R1=(Sgm Y)"; set p99=(Sgm Y)" * (p") * (Sgm X); A26: {i} c= dom f1 by A2,ZFMISC_1:31; A27: X \/ {i} = dom f1 \/ {i} by XBOOLE_1:39 .= dom f1 by A26,XBOOLE_1:12; card(X \/ {i}) = (card X) + card {i} by CARD_2:40,XBOOLE_1:79; then A28: (card X) + 1 = card(X \/ {i}) by CARD_1:30 .= card Seg len f1 by A27,FINSEQ_1:def 3 .= k1+1 by A9,FINSEQ_1:57; A29: {j} c= dom f2 by A7,ZFMISC_1:31; A30: Y \/ {j} = dom f2 \/ {j} by XBOOLE_1:39 .= dom f2 by A29,XBOOLE_1:12; A31: Sgm X is one-to-one by A20,FINSEQ_3:92; then A32: dom((Sgm X)") = X by A21,FUNCT_1:33; then dom((Sgm X)") c= dom f1 by XBOOLE_1:36; then A33: dom((Sgm X)") c= rng p by FUNCT_2:def 3; A34: now let x; assume A35: x in Y; dom f1 = dom p by A2,FUNCT_2:def 1; then A36: x in dom p by A23,A35,XBOOLE_0:def 5; not x in {j} by A35,XBOOLE_0:def 5; then x <> j by TARSKI:def 1; then p.x <> i by A7,A23,A24,A36,FUNCT_2:56; then A37: not p.x in {i} by TARSKI:def 1; dom f1 = rng p by FUNCT_2:def 3; then p.x in dom f1 by A36,FUNCT_1:3; then p.x in X by A37,XBOOLE_0:def 5; hence x in dom((Sgm X)" * p) by A32,A36,FUNCT_1:11; end; now let x; assume A38: x in dom((Sgm X)" * p); then p.x in dom((Sgm X)") by FUNCT_1:11; then p.x in X by A21,A31,FUNCT_1:33; then not p.x in {i} by XBOOLE_0:def 5; then p.x <> i by TARSKI:def 1; then A39: not x in {j} by A24,TARSKI:def 1; x in dom p by A38,FUNCT_1:11; hence x in Y by A23,A39,XBOOLE_0:def 5; end; then dom((Sgm X)" * p) = Y by A34,TARSKI:1; then A40: dom((Sgm X)" * p) = rng(Sgm Y) by A19,FINSEQ_1:def 13; then rng((Sgm X)" * p * Sgm Y) = rng((Sgm X)" * p) by RELAT_1:28 .= rng((Sgm X)") by A33,RELAT_1:28 .= dom(Sgm X) by A31,FUNCT_1:33; then A41: rng p9=Seg k1 by A20,A28,FINSEQ_3:40; card(Y \/ {j}) = (card Y) + card {j} by CARD_2:40,XBOOLE_1:79; then (card Y) + 1 = card(Y \/ {j}) by CARD_1:30 .= card Seg len f2 by A30,FINSEQ_1:def 3 .= card Seg len f1 by A3,Def34 .= k1+1 by A9,FINSEQ_1:57; then dom(Sgm Y) = Seg k1 by A19,FINSEQ_3:40; then A42: dom p9=Seg k1 by A40,RELAT_1:27; A43: dom Del(f1,i) = Seg k1 by A10,FINSEQ_1:def 3; then reconsider p9 as Function of dom Del(f1,i),dom Del(f1,i) by A41,A42, FUNCT_2:1; A44: p9 is onto by A43,A41,FUNCT_2:def 3; Sgm Y is one-to-one by A19,FINSEQ_3:92; then reconsider p9 as Permutation of dom Del(f1,i) by A31,A44; set R7=p9; reconsider R1,R2,R3,R4,R5,R6,R7,p9,p99 as Function; A45: R3=R4~ by A31,FUNCT_1:def 5; A46: Sgm Y is one-to-one & R5=R6~ by A19,FINSEQ_3:92,FUNCT_1:def 5; reconsider R1,R2,R3,R4,R5,R6,R7 as Relation; p9"=R7~ by FUNCT_1:def 5 .=((R6*R3)~)*(R2~) by RELAT_1:35 .=((R3)~*(R6~))*(R2~) by RELAT_1:35 .=((R4~)~*R5)*R1 by A45,A46,FUNCT_1:def 5 .=p99 by RELAT_1:36; hence thesis; end; then reconsider p9 as Permutation of dom Del(f1,i); take p9; A47: Sgm Y is Function of dom Sgm Y, rng Sgm Y by FUNCT_2:1; now let H1,H2 be GroupWithOperators of O,l being Nat ,n; assume A48: l in dom Del(f1,i); set n1=(Sgm Y).n; reconsider n1 as Nat; A49: (Sgm Y)*(p9") = (Sgm Y) * ((Sgm Y)" * ((p") * (Sgm X))) by A25,RELAT_1:36 .= ((Sgm Y) * (Sgm Y)") * ((p") * (Sgm X)) by RELAT_1:36 .= id Y * ((p") * (Sgm X)) by A22,A15,A47,FUNCT_2:29 .= id Y * p" * (Sgm X) by RELAT_1:36 .= (Y|`p")*(Sgm X) by RELAT_1:92; assume A50: n=p9".l; A51: l in dom (p9") by A48,FUNCT_2:def 1; then n in rng (p9") by A50,FUNCT_1:3; then n in dom Del(f1,i); then n in Seg len Del(f2,j) by A5,A9,A10,A8,FINSEQ_1:def 3; then A52: n in dom Del(f2,j) by FINSEQ_1:def 3; set l1=(Sgm X).l; A53: dom Del(f1,i) c= dom Sgm X by RELAT_1:25; then l1 in rng Sgm X by A48,FUNCT_1:3; then A54: l1 in dom f1 by A21,XBOOLE_0:def 5; assume that A55: H1 = Del(f1,i).l and A56: H2 = Del(f2,j).n; reconsider l1 as Nat; A57: H1 = f1.l1 by A48,A55,A53,FUNCT_1:13; A58: dom f1 = rng p by FUNCT_2:def 3; then A59: l1 in dom(p") by A54,FUNCT_1:33; A60: now assume p".l1 in {j}; then A61: p".l1=p".i by A4,TARSKI:def 1; i in dom(p") by A2,A58,FUNCT_1:33; then l1=i by A59,A61,FUNCT_1:def 4; then i in rng Sgm X by A48,A53,FUNCT_1:3; then not i in {i} by A21,XBOOLE_0:def 5; hence contradiction by TARSKI:def 1; end; p".l1 in rng(p") by A59,FUNCT_1:3; then A62: p".l1 in Y by A23,A60,XBOOLE_0:def 5; dom Del(f2,j) c= dom Sgm Y by RELAT_1:25; then A63: H2 = f2.n1 by A56,A52,FUNCT_1:13; n1 = ((Sgm Y)*(p9")).l by A50,A51,FUNCT_1:13 .= (Y|`p").l1 by A48,A53,A49,FUNCT_1:13 .= p".l1 by A54,A62,FUNCT_2:34; hence H1,H2 are_isomorphic by A3,A54,A57,A63,Def34; end; hence thesis by A5,A9,A10,A8,Def34; end; end; theorem Th107: for G1,G2 being GroupWithOperators of O, s1 being CompositionSeries of G1, s2 being CompositionSeries of G2 st s1 is empty & s2 is empty holds s1 is_equivalent_with s2 proof let G1,G2 be GroupWithOperators of O; let s1 be CompositionSeries of G1; let s2 be CompositionSeries of G2; assume A1: s1 is empty; assume A2: s2 is empty; for n st n + 1 = len s1 holds ex p being Permutation of Seg n st for H1 being StableSubgroup of G1, H2 being StableSubgroup of G2, N1 being normal StableSubgroup of H1, N2 being normal StableSubgroup of H2, i,j st 1<=i & i<=n & j=p.i & H1=s1.i & H2=s2.j & N1=s1.(i+1) & N2=s2.(j+1) holds H1./.N1,H2./.N2 are_isomorphic by A1; hence thesis by A1,A2,Def32; end; theorem Th108: for G1,G2 being GroupWithOperators of O, s1 be CompositionSeries of G1, s2 be CompositionSeries of G2 st s1 is not empty & s2 is not empty holds s1 is_equivalent_with s2 iff ex p being Permutation of dom the_series_of_quotients_of s1 st the_series_of_quotients_of s1, the_series_of_quotients_of s2 are_equivalent_under p,O proof let G1,G2 be GroupWithOperators of O; let s1 be CompositionSeries of G1,s2 be CompositionSeries of G2; assume that A1: s1 is not empty and A2: s2 is not empty; set f2 = the_series_of_quotients_of s2; set f1 = the_series_of_quotients_of s1; hereby assume A3: s1 is_equivalent_with s2; then A4: len s1 = len s2 by Def32; per cases; suppose A5: len s1 <= 1; reconsider fs1=f1,fs2=f2 as FinSequence; set p = the Permutation of dom the_series_of_quotients_of s1; reconsider pf=p as Permutation of dom fs1; fs1 = {} by A5,Def33; then A6: for H1,H2 being GroupWithOperators of O, i,j st i in dom fs1 & j=pf" .i & H1=fs1.i & H2=fs2.j holds H1,H2 are_isomorphic; take p; fs2 = {} by A4,A5,Def33; then len f1 = len f2 by A5,Def33; hence the_series_of_quotients_of s1,the_series_of_quotients_of s2 are_equivalent_under p,O by A6,Def34; end; suppose A7: len s1 > 1; set n = len s1 - 1; len s1 - 1 > 1-1 by A7,XREAL_1:9; then n in NAT by INT_1:3; then reconsider n as Nat; n+1 = len s1; then consider p be Permutation of Seg n such that A8: for H1 being StableSubgroup of G1, H2 being StableSubgroup of G2, N1 being normal StableSubgroup of H1, N2 being normal StableSubgroup of H2, i,j st 1<=i & i<=n & j=p.i & H1=s1.i & H2=s2.j & N1=s1.(i+1) & N2=s2.(j+1) holds H1./.N1,H2./.N2 are_isomorphic by A3,Def32; A9: len s1 = len the_series_of_quotients_of s1 + 1 by A7,Def33; then dom the_series_of_quotients_of s1 = Seg n by FINSEQ_1:def 3; then reconsider p9=p" as Permutation of dom the_series_of_quotients_of s1; reconsider fs1=f1,fs2=f2 as FinSequence; A10: len s2 = len the_series_of_quotients_of s2 + 1 by A4,A7,Def33; reconsider pf=p9 as Permutation of dom fs1; take p9; A11: pf" = p by FUNCT_1:43; now let H19,H29 be GroupWithOperators of O; let i,j; set H1=s1.i; set H2=s2.j; set N1=s1.(i+1); set N2=s2.(j+1); assume A12: i in dom fs1; then A13: i in Seg len fs1 by FINSEQ_1:def 3; then A14: 1<=i by FINSEQ_1:1; A15: i<=len fs1 by A13,FINSEQ_1:1; then A16: i+1<=len fs1+1 by XREAL_1:6; 0+i<1+i by XREAL_1:6; then 1<=i+1 by A14,XXREAL_0:2; then i+1 in Seg len s1 by A9,A16; then A17: i+1 in dom s1 by FINSEQ_1:def 3; assume A18: j = pf".i; 0+len fs1<1+len fs1 by XREAL_1:6; then i<=len s1 by A9,A15,XXREAL_0:2; then i in Seg len s1 by A14,FINSEQ_1:1; then A19: i in dom s1 by FINSEQ_1:def 3; then reconsider H1,N1 as Element of the_stable_subgroups_of G1 by A17, FINSEQ_2:11; reconsider H1,N1 as StableSubgroup of G1 by Def11; reconsider N1 as normal StableSubgroup of H1 by A19,A17,Def28; assume that A20: H19=fs1.i and A21: H29=fs2.j; i in dom p by A9,A13,FUNCT_2:def 1; then A22: j in rng p by A11,A18,FUNCT_1:3; then A23: 1<=j by FINSEQ_1:1; A24: j<=len fs2 by A4,A10,A22,FINSEQ_1:1; then A25: j+1<=len fs2+1 by XREAL_1:6; 0+j<1+j by XREAL_1:6; then 1<=j+1 by A23,XXREAL_0:2; then j+1 in Seg len s2 by A10,A25; then A26: j+1 in dom s2 by FINSEQ_1:def 3; 0+len fs2<1+len fs2 by XREAL_1:6; then j<=len s2 by A10,A24,XXREAL_0:2; then j in Seg len s2 by A23,FINSEQ_1:1; then A27: j in dom s2 by FINSEQ_1:def 3; then reconsider H2,N2 as Element of the_stable_subgroups_of G2 by A26, FINSEQ_2:11; reconsider H2,N2 as StableSubgroup of G2 by Def11; reconsider N2 as normal StableSubgroup of H2 by A27,A26,Def28; dom fs1 = Seg n by A9,FINSEQ_1:def 3; then 1<=i & i<=n by A12,FINSEQ_1:1; then A28: H1./.N1,H2./.N2 are_isomorphic by A8,A11,A18; j in Seg len f2 by A4,A10,A22; then j in dom fs2 by FINSEQ_1:def 3; then H2./.N2 = H29 by A4,A7,A21,Def33; hence H19,H29 are_isomorphic by A7,A12,A20,A28,Def33; end; hence the_series_of_quotients_of s1,the_series_of_quotients_of s2 are_equivalent_under p9,O by A4,A9,A10,Def34; end; end; given p be Permutation of dom the_series_of_quotients_of s1 such that A29: the_series_of_quotients_of s1,the_series_of_quotients_of s2 are_equivalent_under p,O; A30: len f1 = len f2 by A29,Def34; per cases; suppose A31: len s1<=1; A32: len s1>=0+1 by A1,NAT_1:13; A33: now let n; set p = the Permutation of Seg n; assume n + 1 = len s1; then n+1=1 by A31,A32,XXREAL_0:1; then A34: n=0; take p; let H1 be StableSubgroup of G1; let H2 be StableSubgroup of G2; let N1 be normal StableSubgroup of H1; let N2 be normal StableSubgroup of H2; let i,j; assume that A35: 1<=i & i<=n and j=p.i; assume that H1=s1.i and H2=s2.j; assume that N1=s1.(i+1) and N2=s2.(j+1); thus H1./.N1,H2./.N2 are_isomorphic by A34,A35; end; A36: f1 = {} by A31,Def33; now assume A37: len s2<>1; len s2>=0+1 by A2,NAT_1:13; then len s2>1 by A37,XXREAL_0:1; then len f2 + 1 > 0 + 1 by Def33; hence contradiction by A30,A36; end; then len s1 = len s2 by A31,A32,XXREAL_0:1; hence thesis by A33,Def32; end; suppose A38: len s1>1; then A39: len s1 = len f1 + 1 by Def33; A40: now assume len s2<=1; then f2 = {} by Def33; then len f2 = 0; hence contradiction by A30,A38,A39; end; A41: now let n; assume A42: n + 1 = len s1; then A43: dom f1 = Seg n by A39,FINSEQ_1:def 3; then reconsider p9=p" as Permutation of Seg n; take p9; let H1 be StableSubgroup of G1; let H2 be StableSubgroup of G2; let N1 be normal StableSubgroup of H1; let N2 be normal StableSubgroup of H2; let i,j; assume 1<=i & i<=n; then A44: i in dom f1 by A43,FINSEQ_1:1; assume A45: j=p9.i; assume that A46: H1=s1.i and A47: H2=s2.j; assume that A48: N1=s1.(i+1) and A49: N2=s2.(j+1); i in dom p9 by A44,FUNCT_2:def 1; then j in rng p9 by A45,FUNCT_1:3; then j in Seg n; then j in dom f2 by A30,A39,A42,FINSEQ_1:def 3; then A50: f2.j = H2./.N2 by A40,A47,A49,Def33; f1.i = H1./.N1 by A38,A44,A46,A48,Def33; hence H1./.N1,H2./.N2 are_isomorphic by A29,A44,A45,A50,Def34; end; len s1 = len s2 by A30,A39,A40,Def33; hence thesis by A41,Def32; end; end; theorem Th109: s1 is_finer_than s2 & s2 is jordan_holder & len s1 > len s2 implies ex i st i in dom the_series_of_quotients_of s1 & for H st H = ( the_series_of_quotients_of s1).i holds H is trivial proof assume A1: s1 is_finer_than s2; assume A2: s2 is jordan_holder; assume A3: len s1 > len s2; then not s1 is strictly_decreasing by A1,A2,Def31; then not for i st i in dom s1 & i+1 in dom s1 for H1,N1 st H1=s1.i & N1=s1.(i +1) holds not H1./.N1 is trivial by Def30; then consider i,H1,N1 such that A4: i in dom s1 and A5: i+1 in dom s1 and A6: H1=s1.i & N1=s1.(i+1) & H1./.N1 is trivial; i+1 in Seg len s1 by A5,FINSEQ_1:def 3; then A7: i+1 <= len s1 by FINSEQ_1:1; 0+1 <= i+1 by XREAL_1:6; then A8: 1 <= len s1 by A7,XXREAL_0:2; per cases; suppose len s1 <= 1; then A9: len s1 = 1 by A8,XXREAL_0:1; now let i; assume i in dom s1; then i in Seg 1 by A9,FINSEQ_1:def 3; then A10: i = 1 by FINSEQ_1:2,TARSKI:def 1; assume A11: i+1 in dom s1; let H1,N1; assume H1=s1.i; assume N1=s1.(i+1); assume H1./.N1 is trivial; 2 in Seg 1 by A9,A10,A11,FINSEQ_1:def 3; hence contradiction by FINSEQ_1:2,TARSKI:def 1; end; then s1 is strictly_decreasing by Def30; hence thesis by A1,A2,A3,Def31; end; suppose A12: len s1 > 1; take i; A13: i+1-1 <= len s1 - 1 by A7,XREAL_1:9; i in Seg len s1 by A4,FINSEQ_1:def 3; then A14: 1 <= i by FINSEQ_1:1; len s1 = len the_series_of_quotients_of s1 + 1 by A12,Def33; then i in Seg len the_series_of_quotients_of s1 by A14,A13,FINSEQ_1:1; hence A15: i in dom the_series_of_quotients_of s1 by FINSEQ_1:def 3; let H; assume H = (the_series_of_quotients_of s1).i; hence thesis by A6,A12,A15,Def33; end; end; :: this is Lm15 of WSIERP_1 Lm40: for k,m being Element of NAT holds k^fs2 & len fs1=a-1 & len fs2=len fs -a proof let a be Element of NAT; let fs be FinSequence; assume A1: a in dom fs; then a>=1 & a<=len fs by FINSEQ_3:25; then reconsider b=len fs-a, d=a-1 as Element of NAT by INT_1:5; len fs=a+b; then consider fs3,fs2 be FinSequence such that A2: len fs3=a and A3: len fs2=b and A4: fs=fs3^fs2 by FINSEQ_2:22; a=d+1; then consider fs1 be FinSequence,v be set such that A5: fs3=fs1^<*v*> by A2,FINSEQ_2:18; A6: len fs1 + 1=d+1 by A2,A5,FINSEQ_2:16; fs3 <> {} by A1,A2,FINSEQ_3:25; then a in dom fs3 by A2,FINSEQ_5:6; then fs3.a=fs.a by A4,FINSEQ_1:def 7; then fs.a=v by A5,A6,FINSEQ_1:42; hence thesis by A3,A4,A5,A6; end; :: this is Lm20 of WSIERP_1 Lm42: for a being Element of NAT, fs,fs1,fs2 being FinSequence, v being set holds a in dom fs & fs=fs1^<*v*>^fs2 & len fs1=a-1 implies Del(fs,a)=fs1^fs2 proof let a be Element of NAT; let fs,fs1,fs2 be FinSequence; let v be set; assume that A1: a in dom fs and A2: fs=fs1^<*v*>^fs2 and A3: len fs1=a-1; A4: len(Del(fs,a))+1=len fs by A1,WSIERP_1:def 1; len fs=len(fs1^<*v*>)+len fs2 by A2,FINSEQ_1:22 .=len fs1 +1 +len fs2 by FINSEQ_2:16 .=a +len fs2 by A3; then len(Del(fs,a)) =len fs2 +len fs1 by A3,A4; then A5: len(fs1^fs2)=len(Del(fs,a)) by FINSEQ_1:22; A6: len<*v*>=1 by FINSEQ_1:39; A7: fs=fs1^(<*v*>^fs2) by A2,FINSEQ_1:32; then len fs=(a-1) + len(<*v*>^fs2) by A3,FINSEQ_1:22; then A8: len(<*v*>^fs2)=len fs -(a-1); now let e be Nat; assume that A9: 1<=e and A10: e<=len Del(fs,a); reconsider e1=e as Element of NAT by ORDINAL1:def 12; now per cases; suppose A11: e=a; then A14: e1>a-1 by Lm40; then A15: e+1>a by XREAL_1:19; then e+1-a>0 by XREAL_1:50; then A16: e+1-a+1>0+1 by XREAL_1:6; A17: e+1>a-1 by A15,XREAL_1:146,XXREAL_0:2; then e+1-(a-1)>0 by XREAL_1:50; then reconsider f=e+1-(a-1) as Element of NAT by INT_1:3; A18: e+1<=len fs by A4,A10,XREAL_1:6; then A19: e+1-(a-1)<=len(<*v*>^fs2) by A8,XREAL_1:9; thus (fs1^fs2).e=fs2.(e-len fs1) by A3,A5,A10,A14,FINSEQ_1:24 .=fs2.(f-1) by A3 .=(<*v*>^fs2).f by A6,A16,A19,FINSEQ_1:24 .=(fs1^(<*v*>^fs2)).(e1+1) by A3,A7,A17,A18,FINSEQ_1:24 .=(Del(fs,a)).e by A1,A7,A13,WSIERP_1:def 1; end; end; hence (fs1^fs2).e=(Del(fs,a)).e; end; hence thesis by A5,FINSEQ_1:14; end; :: this is Lm22 of WSIERP_1 Lm43: for a being Element of NAT, fs1,fs2 being FinSequence holds (a<=len fs1 implies Del(fs1^fs2,a)=Del(fs1,a)^fs2) & (a>=1 implies Del(fs1^fs2,len fs1 +a)= fs1^Del(fs2,a)) proof let a be Element of NAT; let fs1,fs2 be FinSequence; set f=fs1^fs2; A1: len f=len fs1 + len fs2 by FINSEQ_1:22; A2: now set f2=fs1^Del(fs2,a); set f1= Del(f,len fs1 + a); assume A3: a>=1; now per cases; suppose A4: a>len fs2; then A5: not a in dom fs2 by FINSEQ_3:25; len fs1 + a>len f by A1,A4,XREAL_1:6; then not (len fs1 + a) in dom f by FINSEQ_3:25; hence f1=fs1^fs2 by WSIERP_1:def 1 .=f2 by A5,WSIERP_1:def 1; end; suppose A6: a<=len fs2; then A7: a in dom fs2 by A3,FINSEQ_3:25; a-1>=0 by A3,XREAL_1:48; then A8: (a-1)+len fs1>=0+len fs1 by XREAL_1:6; A9: len fs1 + a>=1 by A3,NAT_1:12; len fs1 + a <= len f by A1,A6,XREAL_1:6; then A10: (len fs1 + a) in dom f by A9,FINSEQ_3:25; then consider g1,g2 being FinSequence such that A11: f=g1^<*f.(len fs1 +a)*>^g2 and A12: len g1=len fs1 +a -1 and len g2=len f -(len fs1 +a) by Lm41; A13: f1=g1^g2 by A10,A11,A12,Lm42; f=g1^(<*f.(len fs1 +a)*>^g2) by A11,FINSEQ_1:32; then consider t being FinSequence such that A14: fs1^t=g1 by A12,A8,FINSEQ_1:47; fs1^(t^<*f.(len fs1 +a)*>^g2)=fs1^(t^<*f.(len fs1 +a)*>)^g2 by FINSEQ_1:32 .=f by A11,A14,FINSEQ_1:32; then A15: fs2=t^<*f.(len fs1 +a)*>^g2 by FINSEQ_1:33; len fs1 +(a-1)=len fs1 +len t by A12,A14,FINSEQ_1:22; then Del(fs2,a)=t^g2 by A7,A15,Lm42; hence f1=f2 by A13,A14,FINSEQ_1:32; end; end; hence f1=f2; end; now set f3=<*f.a*>; set f2=((Del(fs1,a))^fs2); set f1= Del(f,a); assume A16: a<=len fs1; len fs1<=len f by A1,NAT_1:11; then A17: a<=len f by A16,XXREAL_0:2; now per cases; suppose A18: a<1; then A19: not a in dom fs1 by FINSEQ_3:25; not a in dom f by A18,FINSEQ_3:25; hence f1=f by WSIERP_1:def 1 .= f2 by A19,WSIERP_1:def 1; end; suppose A20: a>=1; then A21: a in dom f by A17,FINSEQ_3:25; then consider g1,g2 being FinSequence such that A22: f=g1^f3^g2 and A23: len g1=a-1 and len g2=len f -a by Lm41; len(g1^f3)=a-1+1 by A23,FINSEQ_2:16 .=a; then consider t being FinSequence such that A24: fs1=g1^f3^t by A16,A22,FINSEQ_1:47; g1^f3^g2=g1^f3^(t^fs2) by A22,A24,FINSEQ_1:32; then A25: g2=t^fs2 by FINSEQ_1:33; a in dom fs1 by A16,A20,FINSEQ_3:25; then A26: Del(fs1,a)=g1^t by A23,A24,Lm42; thus f1=g1^g2 by A21,A22,A23,Lm42 .=f2 by A26,A25,FINSEQ_1:32; end; end; hence f1=f2; end; hence thesis by A2; end; Lm44: for D being non empty set, f being FinSequence of D, p being Element of D, n being Nat st n in dom f holds f = Del(Ins(f,n,p),n+1) proof let D be non empty set; let f be FinSequence of D; let p be Element of D; let n be Nat; set fs1=f|n^<*p*>; set fs2=(f/^n); assume n in dom f; then n in Seg len f by FINSEQ_1:def 3; then n<= len f by FINSEQ_1:1; then A1: len(f|n) = n by FINSEQ_1:59; len fs1 = len(f|n) + len <*p*> by FINSEQ_1:22 .= n + 1 by A1,FINSEQ_1:39; then Del(Ins(f,n,p),n+1) = Del(fs1,n+1)^fs2 by Lm43 .= f|n^fs2 by A1,WSIERP_1:40; hence thesis by RFINSEQ:8; end; :: ALG I.4.7 Proposition 9 theorem Th110: len s1 > 1 implies (s1 is jordan_holder iff for i st i in dom the_series_of_quotients_of s1 holds (the_series_of_quotients_of s1).i is strict simple GroupWithOperators of O) proof assume A1: len s1 > 1; A2: now assume A3: s1 is jordan_holder; assume not for i st i in dom the_series_of_quotients_of s1 holds ( the_series_of_quotients_of s1).i is strict simple GroupWithOperators of O; then consider i such that A4: i in dom the_series_of_quotients_of s1 and A5: (the_series_of_quotients_of s1).i is not strict simple GroupWithOperators of O; A6: i in Seg len the_series_of_quotients_of s1 by A4,FINSEQ_1:def 3; then A7: i<=len the_series_of_quotients_of s1 by FINSEQ_1:1; len s1 = len the_series_of_quotients_of s1 + 1 by A1,Def33; then A8: i+1 <= len s1 by A7,XREAL_1:6; A9: 0+1<=i+1 by XREAL_1:6; then i+1 in Seg len s1 by A8; then A10: i+1 in dom s1 by FINSEQ_1:def 3; 0+len the_series_of_quotients_of s1< 1+len the_series_of_quotients_of s1 by XREAL_1:6; then A11: len the_series_of_quotients_of s1 < len s1 by A1,Def33; then A12: i<=len s1 by A7,XXREAL_0:2; 1<=i by A6,FINSEQ_1:1; then i in Seg len s1 by A12,FINSEQ_1:1; then A13: i in dom s1 by FINSEQ_1:def 3; then reconsider H1=s1.i,N1=s1.(i+1) as Element of the_stable_subgroups_of G by A10, FINSEQ_2:11; reconsider H1,N1 as strict StableSubgroup of G by Def11; reconsider N1 as strict normal StableSubgroup of H1 by A13,A10,Def28; A14: H1./.N1 is not strict simple GroupWithOperators of O by A1,A4,A5,Def33; per cases by A14,Def13; suppose A15: H1./.N1 is trivial; s1 is strictly_decreasing by A3,Def31; hence contradiction by A13,A10,A15,Def30; end; suppose ex H being strict normal StableSubgroup of H1./.N1 st H <> (Omega).(H1./.N1) & H <> (1).(H1./.N1); then consider H be strict normal StableSubgroup of H1./.N1 such that A16: H <> (Omega).(H1./.N1) and A17: H <> (1).(H1./.N1); N1 = Ker nat_hom N1 by Th48; then consider N2 be strict StableSubgroup of H1 such that A18: the carrier of N2 = (nat_hom N1)"(the carrier of H) and A19: H is normal implies N1 is normal StableSubgroup of N2 & N2 is normal by Th78; A20: N2 is strict StableSubgroup of G by Th11; reconsider i as Element of NAT by ORDINAL1:def 12; A21: 1<=i & s1 is non empty by A1,A6,FINSEQ_1:1; reconsider N2 as Element of the_stable_subgroups_of G by A20,Def11; set s2 = Ins(s1,i,N2); A22: len s2 = len s1 + 1 by FINSEQ_5:69; then A23: s1<>s2; A24: now let j be Nat; assume A25: j in dom s2; then A26: j in Seg len s2 by FINSEQ_1:def 3; then A27: 1<=j by FINSEQ_1:1; A28: j<=len s2 by A26,FINSEQ_1:1; ji by XXREAL_0:1; then j+1<=i or j=i or j>=i+1 by NAT_1:13; then A29: j+1-1<=i-1 or j=i or j>=i+1 by XREAL_1:9; assume A30: j+1 in dom s2; then A31: j+1 in Seg len s2 by FINSEQ_1:def 3; then A32: 1<=j+1 by FINSEQ_1:1; A33: j+1<=len s2 by A31,FINSEQ_1:1; let H19,H29; assume A34: H19=s2.j; assume A35: H29=s2.(j+1); per cases by A29,XXREAL_0:1; suppose A36: j<=i-1; A37: Seg len(s1|i) = Seg i by A11,A7,FINSEQ_1:59,XXREAL_0:2; A38: dom(s1|i) c= dom s1 by RELAT_1:60; -1+i<0+i by XREAL_1:6; then j<=i by A36,XXREAL_0:2; then j in Seg len (s1|i) by A27,A37,FINSEQ_1:1; then A39: j in dom(s1|i) by FINSEQ_1:def 3; j+1<=i-1+1 by A36,XREAL_1:6; then j+1 in Seg len (s1|i) by A32,A37; then A40: j+1 in dom(s1|i) by FINSEQ_1:def 3; A41: s2.(j+1) = s2/.(j+1) by A30,PARTFUN1:def 6 .= s1/.(j+1) by A40,FINSEQ_5:72 .= s1.(j+1) by A38,A40,PARTFUN1:def 6; s2.j = s2/.j by A25,PARTFUN1:def 6 .= s1/.j by A39,FINSEQ_5:72 .= s1.j by A38,A39,PARTFUN1:def 6; hence H29 is normal StableSubgroup of H19 by A34,A35,A38,A39,A40,A41 ,Def28; end; suppose A42: j=i; then A43: j in Seg i by A27; Seg len(s1|i) = Seg i by A11,A7,FINSEQ_1:59,XXREAL_0:2; then A44: j in dom(s1|i) by A43,FINSEQ_1:def 3; A45: dom(s1|i) c= dom s1 by RELAT_1:60; A46: s2.j = s2/.j by A25,PARTFUN1:def 6 .= s1/.j by A44,FINSEQ_5:72 .= s1.j by A45,A44,PARTFUN1:def 6; s2.(j+1) = s2/.(i+1) by A30,A42,PARTFUN1:def 6 .= N2 by A11,A7,FINSEQ_5:73,XXREAL_0:2; hence H29 is normal StableSubgroup of H19 by A19,A34,A35,A42,A46; end; suppose A47: j=i+1; i+1<=len s1+1 by A12,XREAL_1:6; then i+1<=len s2 by FINSEQ_5:69; then i+1 in Seg len s2 by A9; then i+1 in dom s2 by FINSEQ_1:def 3; then A48: H19 = s2/.(i+1) by A34,A47,PARTFUN1:def 6 .= N2 by A11,A7,FINSEQ_5:73,XXREAL_0:2; i+1+1<=len s1+1 by A8,XREAL_1:6; then A49: i+1+1<=len s2 by FINSEQ_5:69; 1<=i+1+1 by A9,XREAL_1:6; then i+1+1 in Seg len s2 by A49; then i+1+1 in dom s2 by FINSEQ_1:def 3; then H29 = s2/.(i+1+1) by A35,A47,PARTFUN1:def 6 .= s1/.(i+1) by A8,FINSEQ_5:74 .= N1 by A10,PARTFUN1:def 6; hence H29 is normal StableSubgroup of H19 by A19,A48; end; suppose A50: i+1i by XXREAL_0:1; then j+1<=i or j=i or j>=i+1 by NAT_1:13; then A67: j+1-1<=i-1 or j=i or j>=i+1 by XREAL_1:9; assume A68: j+1 in dom s2; then A69: j+1 in Seg len s2 by FINSEQ_1:def 3; then A70: 1<=j+1 by FINSEQ_1:1; A71: j+1<=len s2 by A69,FINSEQ_1:1; let H19,N19; assume A72: H19=s2.j; A73: j<=len s2 by A65,FINSEQ_1:1; A74: s1 is strictly_decreasing by A3,Def31; assume A75: N19=s2.(j+1); per cases by A67,XXREAL_0:1; suppose A76: j<=i-1; A77: Seg len(s1|i) = Seg i by A11,A7,FINSEQ_1:59,XXREAL_0:2; A78: dom(s1|i) c= dom s1 by RELAT_1:60; -1+i<0+i by XREAL_1:6; then j<=i by A76,XXREAL_0:2; then j in Seg len (s1|i) by A66,A77,FINSEQ_1:1; then A79: j in dom(s1|i) by FINSEQ_1:def 3; j+1<=i-1+1 by A76,XREAL_1:6; then j+1 in Seg len (s1|i) by A70,A77; then A80: j+1 in dom(s1|i) by FINSEQ_1:def 3; A81: s2.(j+1) = s2/.(j+1) by A68,PARTFUN1:def 6 .= s1/.(j+1) by A80,FINSEQ_5:72 .= s1.(j+1) by A78,A80,PARTFUN1:def 6; s2.j = s2/.j by A64,PARTFUN1:def 6 .= s1/.j by A79,FINSEQ_5:72 .= s1.j by A78,A79,PARTFUN1:def 6; hence H19./.N19 is not trivial by A72,A75,A74,A78,A79,A80,A81,Def30; end; suppose A82: j=i; then A83: j in Seg i by A66; Seg len(s1|i) = Seg i by A11,A7,FINSEQ_1:59,XXREAL_0:2; then A84: j in dom(s1|i) by A83,FINSEQ_1:def 3; A85: s2.(j+1) = s2/.(i+1) by A68,A82,PARTFUN1:def 6 .= N2 by A11,A7,FINSEQ_5:73,XXREAL_0:2; reconsider N2 as normal StableSubgroup of H1 by A19; A86: dom(s1|i) c= dom s1 by RELAT_1:60; A87: s2.j = s2/.j by A64,PARTFUN1:def 6 .= s1/.j by A84,FINSEQ_5:72 .= s1.j by A86,A84,PARTFUN1:def 6; now assume H19./.N19 is trivial; then H1 = N2 by A72,A75,A82,A85,A87,Th76; hence contradiction by A16,A18,Th80; end; hence H19./.N19 is not trivial; end; suppose A88: j=i+1; i+1<=len s1+1 by A12,XREAL_1:6; then i+1<=len s2 by FINSEQ_5:69; then i+1 in Seg len s2 by A9; then i+1 in dom s2 by FINSEQ_1:def 3; then A89: H19 = s2/.(i+1) by A72,A88,PARTFUN1:def 6 .= N2 by A11,A7,FINSEQ_5:73,XXREAL_0:2; i+1+1<=len s1+1 by A8,XREAL_1:6; then A90: i+1+1<=len s2 by FINSEQ_5:69; 1<=i+1+1 by A9,XREAL_1:6; then i+1+1 in Seg len s2 by A90; then i+1+1 in dom s2 by FINSEQ_1:def 3; then A91: N19 = s2/.(i+1+1) by A75,A88,PARTFUN1:def 6 .= s1/.(i+1) by A8,FINSEQ_5:74 .= N1 by A10,PARTFUN1:def 6; now assume H19./.N19 is trivial; then the carrier of N1 = (nat_hom N1)"(the carrier of H) by A18,A89 ,A91,Th76; hence contradiction by A17,Th81; end; hence H19./.N19 is not trivial; end; suppose A92: i+1 1 by NAT_1:13; then len the_series_of_quotients_of s1 + 1 = len s1 by Def33; then len the_series_of_quotients_of s1 = len s1 - 1; then i+1-1 <= len the_series_of_quotients_of s1 by A109,XREAL_1:9; then i in Seg len the_series_of_quotients_of s1 by A110,FINSEQ_1:1; then A112: i in dom the_series_of_quotients_of s1 by FINSEQ_1:def 3; then (the_series_of_quotients_of s1).i=H1./.N1 by A107,A111,Def33; then H1./.N1 is strict simple GroupWithOperators of O by A103,A112; hence contradiction by A108,Def13; end; suppose ex s2 st s2<>s1 & s2 is strictly_decreasing & s2 is_finer_than s1; then consider s2 such that A113: s2<>s1 and A114: s2 is strictly_decreasing and A115: s2 is_finer_than s1; consider i,j such that A116: i in dom s1 and A117: i in dom s2 and A118: i+1 in dom s1 and A119: i+1 in dom s2 and A120: j in dom s2 & i+1s2.(i+1) and A123: s1.(i+1)=s2.j by A1,A113,A114,A115,Th100; reconsider H1=s1.i,H2=s1.(i+1),H=s2.(i+1) as Element of the_stable_subgroups_of G by A116,A118,A119,FINSEQ_2:11; reconsider H1,H2,H as strict StableSubgroup of G by Def11; reconsider H2 as strict normal StableSubgroup of H1 by A116,A118,Def28; reconsider H as strict normal StableSubgroup of H1 by A117,A119,A121 ,Def28; reconsider H29=H2 as normal StableSubgroup of H by A119,A120,A123,Th40 ,Th101; reconsider J = H./.H29 as strict normal StableSubgroup of H1./.H2 by Th44 ; A124: now assume J = (Omega).(H1./.H2); then A125: the carrier of H = union Cosets H2 by Th22; then A126: H = H1 by Lm5,Th22; then reconsider H1 as strict normal StableSubgroup of H; H1 = (Omega).H by A125,Lm5,Th22; then H./.H1 is trivial by Th57; hence contradiction by A114,A117,A119,A121,A126,Def30; end; reconsider H3 = the HGrWOpStr of H2 as strict normal StableSubgroup of H by A119,A120,A123,Th40,Th101; now assume J = (1).(H1./.H2); then union Cosets H3 = union {1_(H1./.H2)} by Def8; then the carrier of H = union {1_(H1./.H2)} by Th22; then the carrier of H = 1_(H1./.H2) by ZFMISC_1:25; then the carrier of H = carr H2 by Th43; hence contradiction by A122,Lm5; end; then A127: H1./.H2 is not simple GroupWithOperators of O by A124,Def13; i+1 in Seg len s1 by A118,FINSEQ_1:def 3; then A128: i+1 <= len s1 by FINSEQ_1:1; i in Seg len s1 by A116,FINSEQ_1:def 3; then A129: 1 <= i by FINSEQ_1:1; then 1+1 <= i+1 by XREAL_1:6; then 1+1 <= len s1 by A128,XXREAL_0:2; then A130: len s1 > 1 by NAT_1:13; then len the_series_of_quotients_of s1 + 1 = len s1 by Def33; then len the_series_of_quotients_of s1 = len s1 - 1; then i+1-1 <= len the_series_of_quotients_of s1 by A128,XREAL_1:9; then i in Seg len the_series_of_quotients_of s1 by A129,FINSEQ_1:1; then A131: i in dom the_series_of_quotients_of s1 by FINSEQ_1:def 3; then (the_series_of_quotients_of s1).i=H1./.H2 by A130,Def33; hence contradiction by A103,A127,A131; end; end; hence thesis by A2; end; theorem Th111: 1<=i & i<=len s1-1 implies s1.i is strict StableSubgroup of G & s1.(i+1) is strict StableSubgroup of G proof assume that A1: 1<=i and A2: i<=len s1-1; A3: 0+i<=1+i by XREAL_1:6; A4: i+1<=len s1-1+1 by A2,XREAL_1:6; then i<=len s1 by A3,XXREAL_0:2; then i in Seg len s1 by A1,FINSEQ_1:1; then i in dom s1 by FINSEQ_1:def 3; then s1.i is Element of the_stable_subgroups_of G by FINSEQ_2:11; hence s1.i is strict StableSubgroup of G by Def11; 1<=i+1 by A1,A3,XXREAL_0:2; then i+1 in Seg len s1 by A4; then i+1 in dom s1 by FINSEQ_1:def 3; then s1.(i+1) is Element of the_stable_subgroups_of G by FINSEQ_2:11; hence thesis by Def11; end; theorem Th112: 1<=i & i<=len s1-1 & H1=s1.i & H2=s1.(i+1) implies H2 is normal StableSubgroup of H1 proof assume that A1: 1<=i and A2: i<=len s1-1; A3: i+1<=len s1-1+1 by A2,XREAL_1:6; A4: 0+i<=1+i by XREAL_1:6; then 1<=i+1 by A1,XXREAL_0:2; then i+1 in Seg len s1 by A3; then A5: i+1 in dom s1 by FINSEQ_1:def 3; i<=len s1 by A4,A3,XXREAL_0:2; then i in Seg len s1 by A1,FINSEQ_1:1; then A6: i in dom s1 by FINSEQ_1:def 3; assume H1=s1.i & H2=s1.(i+1); hence thesis by A5,A6,Def28; end; theorem Th113: s1 is_equivalent_with s1 proof per cases; suppose s1 is empty; hence thesis by Th107; end; suppose A1: s1 is not empty; set f1=the_series_of_quotients_of s1; now set p = id dom f1; reconsider p as Function of dom f1,dom f1; rng p = dom f1 by RELAT_1:45; then p is onto by FUNCT_2:def 3; then reconsider p as Permutation of dom f1; take p; A2: now let H1,H2 be GroupWithOperators of O; let i,j; assume A3: i in dom f1 & j=p".i; A4: p" = p by FUNCT_1:45; assume H1=f1.i & H2=f1.j; hence H1,H2 are_isomorphic by A3,A4,FUNCT_1:18; end; len f1 = len f1; hence f1,f1 are_equivalent_under p,O by A2,Def34; end; hence thesis by A1,Th108; end; end; theorem Th114: (len s1<=1 or len s2<=1) & len s1<=len s2 implies s2 is_finer_than s1 proof assume A1: len s1<=1 or len s2<=1; assume A2: len s1<=len s2; then A3: len s1 <=1 by A1,XXREAL_0:2; per cases; suppose A4: len s1=1; then A5: s1 = <* s1.1 *> by FINSEQ_1:40; now reconsider D=Seg len s2 as non empty set by A2,A4; set x={1}; take x; set f=s2; set p=<*1*>; dom f = Seg len s2 & rng f c= the_stable_subgroups_of G by FINSEQ_1:def 3 ; then reconsider f as Function of D, the_stable_subgroups_of G by FUNCT_2:2; A6: 1 in Seg len s2 by A2,A4; then 1 in dom s2 by FINSEQ_1:def 3; hence x c= dom s2 by ZFMISC_1:31; {1} c= D by A6,ZFMISC_1:31; then rng p c= D by FINSEQ_1:38; then reconsider p as FinSequence of D by FINSEQ_1:def 4; Sgm x = p & f * p = <* f.1 *> by FINSEQ_2:35,FINSEQ_3:44; then s2 * Sgm x = <* (Omega).G *> by Def28; hence s1 = s2 * Sgm x by A5,Def28; end; hence thesis by Def29; end; suppose len s1<>1; then len s1<0+1 by A3,XXREAL_0:1; then A7: s1={} by NAT_1:13; now set x={}; take x; thus x c= dom s2 by XBOOLE_1:2; thus s1 = s2 * Sgm x by A7,FINSEQ_3:43; end; hence thesis by Def29; end; end; theorem Th115: s1 is_equivalent_with s2 & s1 is jordan_holder implies s2 is jordan_holder proof assume A1: s1 is_equivalent_with s2; assume A2: s1 is jordan_holder; per cases; suppose A3: len s1<=0+1; per cases by A3,NAT_1:25; suppose A4: len s1=0; then len s2=0 by A1,Def32; then A5: s2 = {}; s1={} by A4; hence thesis by A2,A5; end; suppose A6: len s1=1; then A7: s1.1=(1).G by Def28; A8: len s2=1 by A1,A6,Def32; s1 = <* s1.1 *> by A6,FINSEQ_1:40 .= <* s2.1 *> by A7,A8,Def28 .= s2 by A8,FINSEQ_1:40; hence thesis by A2; end; end; suppose A9: len s1>1; set f2 = the_series_of_quotients_of s2; set f1 = the_series_of_quotients_of s1; A10: s1 is not empty by A9; A11: len s2>1 by A1,A9,Def32; then s2 is not empty; then consider p be Permutation of dom the_series_of_quotients_of s1 such that A12: the_series_of_quotients_of s1,the_series_of_quotients_of s2 are_equivalent_under p,O by A1,A10,Th108; A13: len f1 = len f2 by A12,Def34; now let j; set i=p.j; set H1=f1.i; set H2=f2.j; assume A14: j in dom f2; then A15: f2.j in rng f2 by FUNCT_1:3; A16: dom f1 = Seg len f1 by FINSEQ_1:def 3 .= Seg len f2 by A12,Def34 .= dom f2 by FINSEQ_1:def 3; then A17: p.j in dom f2 by A14,FUNCT_2:5; then reconsider i as Element of NAT; p.j in Seg len f2 by A17,FINSEQ_1:def 3; then A18: i in dom f1 by A13,FINSEQ_1:def 3; then f1.i in rng f1 by FUNCT_1:3; then reconsider H1,H2 as strict GroupWithOperators of O by A15,Th102; i in dom f1 & j=p".i by A14,A16,FUNCT_2:5,26; then A19: H1,H2 are_isomorphic by A12,Def34; H1 is strict simple GroupWithOperators of O by A2,A9,A18,Th110; hence f2.j is strict simple GroupWithOperators of O by A19,Th82; end; hence thesis by A11,Th110; end; end; Lm45: for k,l being Nat st k in Seg l & len s1>1 & len s2>1 & l=(len s1-1)*( len s2-1)+1 holds k=(len s1-1)*(len s2-1)+1 or ex i,j st k=(i-1)*(len s2-1)+j & 1<=i & i<=len s1-1 & 1<=j & j<=len s2-1 proof let k,l be Nat; set l9 = len s1-1; set l99 = len s2-1; assume A1: k in Seg l; then A2: k<=l by FINSEQ_1:1; assume that A3: len s1>1 and A4: len s2>1 and A5: l=(len s1-1)*(len s2-1)+1; assume not k = (len s1-1)*(len s2-1)+1; then A6: k1+1 by A4,XREAL_1:6; then len s2>=2 by NAT_1:13; then A7: len s2-1>=2-1 by XREAL_1:9; len s1-1>1-1 by A3,XREAL_1:9; then reconsider l9 as Element of NAT by INT_1:3; A8: len s2-1>1-1 by A4,XREAL_1:9; then reconsider l99 as Element of NAT by INT_1:3; A9: k = (k div l99)*l99 + (k mod l99) by A8,NAT_D:2; per cases; suppose A10: k mod l99=0; set i=k div l99; set j=l99; take i,j; thus k = (i-1)*(len s2-1)+j by A9,A10; i>0 by A1,A9,A10,FINSEQ_1:1; then i+1>0+1 by XREAL_1:6; hence 1<=i by NAT_1:13; i*l99<=(len s1-1)*l99 by A5,A6,A9,A10,INT_1:7; then i*l99/l99<=(len s1-1)*l99/l99 by XREAL_1:72; then i<=(len s1-1)*l99/l99 by A8,XCMPLX_1:89; hence i<=len s1-1 by A8,XCMPLX_1:89; thus thesis by A7; end; suppose A11: k mod l99<>0; set i=k div l99+1; set j=k mod l99; take i,j; thus k = (i-1)*(len s2-1)+j by A8,NAT_D:2; 0+1<=k div l99+1 by XREAL_1:6; hence 1<=i; k+1<=l by A6,INT_1:7; then A12: k+1-1<=l-1 by XREAL_1:9; k mod l99 + l99*(k div l99)>=0+l99*(k div l99) by XREAL_1:6; then A13: (k div l99)*l99<=k by A8,NAT_D:2; k<>l9*l99 by A11,NAT_D:13; then k<(len s1-1)*l99 by A5,A12,XXREAL_0:1; then (k div l99)*l99<(len s1-1)*l99 by A13,XXREAL_0:2; then (k div l99)*l99/l99<(len s1-1)*l99/l99 by A8,XREAL_1:74; then k div l99<(len s1-1)*l99/l99 by A8,XCMPLX_1:89; then k div l990+1 by A11,XREAL_1:6; hence 1<=j by NAT_1:13; thus thesis by A8,NAT_D:1; end; end; Lm46: for i1,j1,i2,j2 being Nat, s1,s2 st len s2>1 & (i1-1)*(len s2-1)+j1=(i2- 1)*(len s2-1)+j2 & 1<=i1 & 1<=j1 & j1<=len s2-1 & 1<=i2 & 1<=j2 & j2<=len s2-1 holds j1=j2 & i1=i2 proof let i1,j1,i2,j2 be Nat; let s1,s2; set l99 = len s2-1; set i19 = i1-1; set i29 = i2-1; assume len s2>1; then A1: len s2-1>1-1 by XREAL_1:9; then reconsider l99 as Element of NAT by INT_1:3; A2: l99/l99=1 by A1,XCMPLX_1:60; assume A3: (i1-1)*(len s2-1)+j1=(i2-1)*(len s2-1)+j2; assume that A4: 1<=i1 and A5: 1<=j1 and A6: j1<=len s2-1; i1-1>=1-1 by A4,XREAL_1:9; then reconsider i19 as Element of NAT by INT_1:3; assume that A7: 1<=i2 and A8: 1<=j2 and A9: j2<=len s2-1; i2-1>=1-1 by A7,XREAL_1:9; then reconsider i29 as Element of NAT by INT_1:3; A10: j1 mod l99 = (i19*l99+j1) mod l99 by NAT_D:21 .= (i29*l99+j2) mod l99 by A3 .= j2 mod l99 by NAT_D:21; A11: j1=j2 proof per cases; suppose A12: j1=l99; assume j2<>j1; then j2l99; then j1l99; then j21 & k=(i-1)*( len s2-1)+j & 1<=i & i<=len s1-1 & 1<=j & j<=len s2-1 holds 1<=k & k<=(len s1-1 )*(len s2-1) proof let k be integer number; let i,j be Nat; let s1,s2; set l9=len s1-1; set l99=len s2-1; assume len s2>1; then A1: len s2-1>1-1 by XREAL_1:9; assume A2: k=(i-1)*(len s2-1)+j; assume that A3: 1<=i and A4: i<=len s1-1; assume that A5: 1<=j and A6: j<=len s2-1; i-1<=l9-1 by A4,XREAL_1:9; then (i-1)*l99<=(l9-1)*l99 by A1,XREAL_1:64; then A7: k<=l9*l99-1*l99+l99 by A2,A6,XREAL_1:7; 1-1<=i-1 by A3,XREAL_1:9; then 0+1<=(i-1)*(len s2-1)+j by A5,A1,XREAL_1:7; hence thesis by A2,A7; end; begin :: The Schreier Refinement Theorem definition let O,G,s1,s2; assume that A1: len s1>1 and A2: len s2>1; func the_schreier_series_of(s1,s2) -> CompositionSeries of G means :Def35: for k,i,j being Nat, H1,H2,H3 being StableSubgroup of G holds (k=(i-1)*(len s2- 1)+j & 1<=i & i<=len s1-1 & 1<=j & j<=len s2-1 & H1=s1.(i+1) & H2=s1.i & H3=s2. j implies it.k = H1"\/"(H2/\H3)) & (k=(len s1-1)*(len s2-1) + 1 implies it.k = (1).G) & len it = (len s1-1)*(len s2-1) + 1; existence proof len s2-1>1-1 by A2,XREAL_1:9; then reconsider l99 = len s2-1 as Element of NAT by INT_1:3; len s2+1>1+1 by A2,XREAL_1:6; then len s2>=2 by NAT_1:13; then A3: len s2-1>=2-1 by XREAL_1:9; len s1-1>1-1 by A1,XREAL_1:9; then reconsider l9 = len s1-1 as Element of NAT by INT_1:3; defpred P[set,set] means for i,j being Nat, H1,H2,H3 being StableSubgroup of G holds ($1=(i-1)*(len s2-1)+j & 1<=i & i<=len s1-1 & 1<=j & j<=len s2-1 & H1=s1.(i+1) & H2=s1.i & H3=s2.j implies $2 = H1"\/"(H2/\H3)) & ($1=(len s1-1)*( len s2-1) + 1 implies $2 = (1).G); len s2-1>1-1 by A2,XREAL_1:9; then A4: l99/l99=1 by XCMPLX_1:60; len s1+1>1+1 by A1,XREAL_1:6; then len s1>=2 by NAT_1:13; then A5: len s1-1>=2-1 by XREAL_1:9; then A6: (len s1-1)*(len s2-1)+1>=0+1 by A3,XREAL_1:6; reconsider l=(len s1-1)*(len s2-1)+1 as Element of NAT by A5,A3,INT_1:3; A7: 1 in Seg l by A6; A8: for k being Nat st k=(len s1-1)*(len s2-1)+1 holds not ex i,j st k=(i -1)*(len s2-1)+j & 1<=i & i<=len s1-1 & 1<=j & j<=len s2-1 proof let k be Nat; assume A9: k=(len s1-1)*(len s2-1)+1; assume ex i,j st k=(i-1)*(len s2-1)+j & 1<=i & i<=len s1-1 & 1<=j & j <=len s2-1; then consider i,j such that A10: k=(i-1)*(len s2-1)+j and A11: 1<=i and A12: i<=len s1-1 and A13: 1<=j and A14: j<=len s2-1; set i9 = i-1; i-1>=1-1 by A11,XREAL_1:9; then reconsider i9 as Element of NAT by INT_1:3; A15: 1 mod l99 = (l9*l99+1) mod l99 by NAT_D:21 .= (i9*l99+j) mod l99 by A9,A10 .= j mod l99 by NAT_D:21; j=1 proof per cases; suppose A16: j=l99; assume j<>1; then 1l99; then jl99; 1<=l99 by A13,A14,XXREAL_0:2; then 1l by NAT_1:13; then consider i,j such that A49: k=(i-1)*(len s2-1)+j and A50: 1<=i and A51: i<=len s1-1 and A52: 1<=j and A53: j<=len s2-1 by A1,A2,A33,A47,Lm45; reconsider H19=s1.(i+1),H29=s1.i,H39=s2.j as strict StableSubgroup of G by A50,A51,A52,A53,Th111; A54: f.k = H19"\/"(H29/\H39) by A33,A47,A49,A50,A51,A52,A53; let H1,H2; assume A55: H1 = f.k; A56: H19 is normal StableSubgroup of H29 by A39,A50,A51; assume A57: H2 = f.(k+1); per cases; suppose A58: j<>len s2-1; reconsider j9=j+1 as Nat; jlen s1-1; set i9=i+1; A64: 0+i9<=1+i9 by XREAL_1:6; set j9=1; H19 is StableSubgroup of H1 by A55,A54,Th35; then H19 is Subgroup of H1 by Def7; then A65: the carrier of H19 c= the carrier of H1 by GROUP_2:def 5; 1+1<=i+1 by A50,XREAL_1:6; then A66: 1<=i9 by XXREAL_0:2; i1-1 by A2,XREAL_1:9; then A70: l99>=0+1 by NAT_1:13; then reconsider H199=s1.(i9+1),H299=s1.i9,H399=s2.j9 as strict StableSubgroup of G by A67,A66,Th111; 1<=i9+1 by A66,A64,XXREAL_0:2; then i9+1 in Seg len s1 by A68; then i9+1 in dom s1 by FINSEQ_1:def 3; then A71: H199 is normal StableSubgroup of H299 by A69,Def28; now let x be set; H299 is Subgroup of G by Def7; then A72: the carrier of H299 c= the carrier of G by GROUP_2:def 5; assume x in the carrier of H299; hence x in the carrier of (Omega).G by A72; end; then the carrier of H299 c= the carrier of (Omega).G by TARSKI:def 3; then A73: the carrier of H299 = (the carrier of H299) /\ (the carrier of (Omega).G) by XBOOLE_1:28; A74: H399 = (Omega).G by Def28; k9=(i9-1)*(len s2-1)+j9 by A49,A62; then H2 = H199"\/"(H299/\H399) by A33,A48,A57,A67,A66,A70; then H2 = H199"\/"H299 by A74,A73,Th18; then A75: H2 = H19 by A71,Th36; H29/\H39 is StableSubgroup of H29 by Lm34; then A76: H1 is StableSubgroup of H29 by A55,A56,A54,Th37; then A77: H1 is Subgroup of H29 by Def7; now let H9 be strict Subgroup of H1; assume A78: H9 = the multMagma of H2; now let a be Element of H1; reconsider a9=a as Element of H29 by A76,Th2; now reconsider H1s9=the multMagma of H19 as normal Subgroup of H29 by A56,Lm7; let x be set; assume x in a * H9; then consider b be Element of H1 such that A79: x = a * b and A80: b in H9 by GROUP_2:103; set b9=b; A81: H1 is Subgroup of H29 by A76,Def7; then reconsider b9 as Element of H29 by GROUP_2:42; x = a9 * b9 by A79,A81,GROUP_2:43; then a9 * H1s9 c= H1s9 * a9 & x in a9 * H1s9 by A75,A78,A80, GROUP_2:103,GROUP_3:118; then consider b99 be Element of H29 such that A82: x = b99 * a9 and A83: b99 in H1s9 by GROUP_2:104; b99 in the carrier of H19 by A83,STRUCT_0:def 5; then reconsider b99 as Element of H1 by A65; x = b99 * a by A77,A82,GROUP_2:43; hence x in H9 * a by A75,A78,A83,GROUP_2:104; end; hence a * H9 c= H9 * a by TARSKI:def 3; end; hence H9 is normal by GROUP_3:118; end; hence thesis by A55,A54,A75,Def10,Th35; end; suppose i=len s1-1; then H2 = (1).G by A33,A48,A49,A57,A62 .= (1).H1 by Th15; hence thesis; end; end; end; len s1-1>1-1 & len s2-1>1-1 by A1,A2,XREAL_1:9; then (len s1-1)*(len s2-1)>0*(len s2-1) by XREAL_1:68; then 1<>l; then consider i,j such that A84: 1=(i-1)*(len s2-1)+j and A85: 1<=i and A86: i<=len s1-1 and A87: 1<=j and A88: j<=len s2-1 by A1,A2,A7,Lm45; set i9 = i-1; i-1>=1-1 by A85,XREAL_1:9; then reconsider i9 as Element of NAT by INT_1:3; reconsider H1=s1.(i+1),H2=s1.i,H3=s2.j as StableSubgroup of G by A85,A86 ,A87,A88,Th111; 1 mod l99=(i9*l99+j) mod l99 by A84; then A89: 1 mod l99=j mod l99 by NAT_D:21; A90: j=1 proof per cases; suppose l99=1; hence thesis by A87,A88,XXREAL_0:1; end; suppose l99<>1; then 1l99 by NAT_D:25; then l99>j by A88,XXREAL_0:1; hence thesis by A91,NAT_D:24; end; end; then A92: H3=(Omega).G by Def28; i9*l99/l99=0/l99 by A84,A90; then i9*1=0 by A4,XCMPLX_1:74; then A93: H2=(Omega).G by Def28; f.1 = H1"\/"(H2/\H3) by A33,A7,A84,A85,A86,A87,A88; then f.1 = H1"\/"(Omega).G by A93,A92,Th19; then f.1=(Omega).G by Th34; then reconsider f as CompositionSeries of G by A38,A46,Def28; take f; let k,i,j be Nat, H1,H2,H3 be StableSubgroup of G; A94: for k,i,j being Nat st k=(i-1)*(len s2-1)+j & 1<=i & i<=len s1-1 & 1 <=j & j<=len s2-1 holds k in Seg l proof let k,i,j be Nat; assume A95: k=(i-1)*(len s2-1)+j; assume that A96: 1<=i and A97: i<=len s1-1; assume that A98: 1<=j and A99: j<=len s2-1; i-1<=l9-1 by A97,XREAL_1:9; then (i-1)*l99<=(l9-1)*l99 by XREAL_1:64; then 0+l9*l99<=1+l9*l99 & k<=l9*l99-1*l99+l99 by A95,A99,XREAL_1:7; then A100: k<=(len s1-1)*(len s2-1)+1 by XXREAL_0:2; 1-1<=i-1 by A96,XREAL_1:9; then 0+1<=(i-1)*(len s2-1)+j by A3,A98,XREAL_1:7; hence thesis by A95,A100,FINSEQ_1:1; end; now assume that A101: k=(i-1)*(len s2-1)+j & 1<=i & i<=len s1-1 & 1<=j & j<=len s2-1 and A102: H1=s1.(i+1) & H2=s1.i & H3=s2.j; k in Seg l by A94,A101; hence f.k = H1"\/"(H2/\H3) by A33,A101,A102; end; hence k=(i-1)*(len s2-1)+j & 1<=i & i<=len s1-1 & 1<=j & j<=len s2-1 & H1= s1.(i+1) & H2=s1.i & H3=s2.j implies f.k = H1"\/"(H2/\H3); now assume A103: k=(len s1-1)*(len s2-1) + 1; then k in Seg l by A6; hence f.k = (1).G by A33,A103; end; hence thesis by A33,FINSEQ_1:def 3; end; uniqueness proof let f1,f2 be CompositionSeries of G; assume A104: for k,i,j being Nat, H1,H2,H3 being StableSubgroup of G holds (k =(i-1)*(len s2-1)+j & 1<=i & i<=len s1-1 & 1<=j & j<=len s2-1 & H1=s1.(i+1) & H2=s1.i & H3=s2.j implies f1.k = H1"\/"(H2/\H3)) & (k=(len s1-1)*(len s2-1) + 1 implies f1.k = (1).G) & len f1 = (len s1-1)*(len s2-1) + 1; assume A105: for k,i,j being Nat, H1,H2,H3 being StableSubgroup of G holds (k =(i-1)*(len s2-1)+j & 1<=i & i<=len s1-1 & 1<=j & j<=len s2-1 & H1=s1.(i+1) & H2=s1.i & H3=s2.j implies f2.k = H1"\/"(H2/\H3)) & (k=(len s1-1)*(len s2-1) + 1 implies f2.k = (1).G) & len f2 = (len s1-1)*(len s2-1) + 1; A106: now set l=len f1; let k be Nat; assume k in dom f1; then A107: k in Seg l by FINSEQ_1:def 3; per cases by A1,A2,A104,A107,Lm45; suppose ex i,j st k=(i-1)*(len s2-1)+j & 1<=i & i<=len s1-1 & 1<=j & j<=len s2-1; then consider i,j such that A108: k=(i-1)*(len s2-1)+j and A109: 1<=i & i<=len s1-1 & 1<=j & j<=len s2-1; reconsider H1=s1.(i+1),H2=s1.i,H3=s2.j as StableSubgroup of G by A109 ,Th111; f1.k=H1"\/"(H2/\H3) by A104,A108,A109; hence f1.k = f2.k by A105,A108,A109; end; suppose A110: k=(len s1-1)*(len s2-1)+1; then f1.k=(1).G by A104; hence f1.k = f2.k by A105,A110; end; end; dom f1 = Seg len f2 by A104,A105,FINSEQ_1:def 3 .= dom f2 by FINSEQ_1:def 3; hence thesis by A106,FINSEQ_1:13; end; end; theorem Th116: len s1>1 & len s2>1 implies the_schreier_series_of(s1,s2) is_finer_than s1 proof assume that A1: len s1>1 and A2: len s2>1; now set rR=rng s1; set R=s1; set l=(len s1-1)*(len s2-1) + 1; set X = Seg len s1; set g={[k,(k-1)*(len s2-1)+1] where k is Element of NAT:1<=k & k<=len s1}; now let x; assume x in g; then consider k be Element of NAT such that A3: [k,(k-1)*(len s2-1)+1] = x and 1<=k and k<=len s1; set z=(k-1)*(len s2-1)+1; set y=k; reconsider y,z as set; take y,z; thus x = [y,z] by A3; end; then reconsider g as Relation by RELAT_1:def 1; A4: now let y; assume y in rng g; then consider x such that A5: [x,y] in g by XTUPLE_0:def 13; consider k be Element of NAT such that A6: [k,(k-1)*(len s2-1)+1] = [x,y] and 1<=k and k<=len s1 by A5; (k-1)*(len s2-1)+1=y by A6,XTUPLE_0:1; hence y in REAL by XREAL_0:def 1; end; A7: now let x,y1,y2 be set; assume [x,y1] in g; then consider k be Element of NAT such that A8: [k,(k-1)*(len s2-1)+1] = [x,y1] and 1<=k and k<=len s1; A9: k=x by A8,XTUPLE_0:1; assume [x,y2] in g; then consider k9 be Element of NAT such that A10: [k9,(k9-1)*(len s2-1)+1] = [x,y2] and 1<=k9 and k9<=len s1; k9=x by A10,XTUPLE_0:1; hence y1=y2 by A8,A10,A9,XTUPLE_0:1; end; now let x; assume x in dom g; then consider y such that A11: [x,y] in g by XTUPLE_0:def 12; consider k be Element of NAT such that A12: [k,(k-1)*(len s2-1)+1] = [x,y] and 1<=k and k<=len s1 by A11; k=x by A12,XTUPLE_0:1; hence x in NAT; end; then A13: dom g c= NAT by TARSKI:def 3; reconsider g as Function by A7,FUNCT_1:def 1; A14: rng g c= REAL by A4,TARSKI:def 3; reconsider f=g as PartFunc of dom g, rng g by RELSET_1:4; dom g c= REAL by A13,XBOOLE_1:1; then reconsider f as PartFunc of REAL,REAL by A14,RELSET_1:7; set dR=dom s1; set t=the_schreier_series_of(s1,s2); set fX = f.:X; take fX; reconsider R as Relation of dR,rR by FUNCT_2:1; A15: (id dR)*R = R by FUNCT_2:17; len s2+1>1+1 by A2,XREAL_1:6; then len s2>=2 by NAT_1:13; then A16: len s2-1>=2-1 by XREAL_1:9; len s1+1>1+1 by A1,XREAL_1:6; then len s1>=2 by NAT_1:13; then len s1-1>=2-1 by XREAL_1:9; then reconsider l as Element of NAT by A16,INT_1:3; A17: len the_schreier_series_of(s1,s2) = l by A1,A2,Def35; then A18: dom the_schreier_series_of(s1,s2) = Seg l by FINSEQ_1:def 3; len s2+1>1+1 by A2,XREAL_1:6; then len s2>=2 by NAT_1:13; then A19: len s2-1>=2-1 by XREAL_1:9; now let y be set; assume y in fX; then consider x such that A20: [x,y] in g and x in X by RELAT_1:def 13; consider k be Element of NAT such that A21: [k,(k-1)*(len s2-1)+1]=[x,y] and A22: 1<=k and A23: k<=len s1 by A20; reconsider y9=y as integer number by A21,XTUPLE_0:1; A24: k-1>=1-1 by A22,XREAL_1:9; then A25: y9>0 by A19,A21,XTUPLE_0:1; k-1<=len s1-1 by A23,XREAL_1:9; then A26: (k-1)*(len s2-1)<=(len s1-1)*(len s2-1) by A19,XREAL_1:64; (k-1)*(len s2-1)+1>=0+1 by A19,A24,XREAL_1:6; then A27: y9>=1 by A21,XTUPLE_0:1; reconsider y9 as Element of NAT by A25,INT_1:3; (k-1)*(len s2-1)+1=y by A21,XTUPLE_0:1; then y9<=l by A26,XREAL_1:6; hence y in Seg l by A27; end; then A28: fX c= Seg l by TARSKI:def 3; hence fX c= dom the_schreier_series_of(s1,s2) by A17,FINSEQ_1:def 3; now let x; assume A29: x in X; then reconsider k=x as Element of NAT; set y=(k-1)*(len s2-1)+1; 1<=k & k<=len s1 by A29,FINSEQ_1:1; then [x,y] in f; hence x in dom f by XTUPLE_0:def 12; end; then A30: X c= dom f by TARSKI:def 3; then A31: dom s1 c= dom f by FINSEQ_1:def 3; now let x; assume x in dom f; then consider y such that A32: [x,y] in f by XTUPLE_0:def 12; consider k be Element of NAT such that A33: [k,(k-1)*(len s2-1)+1] = [x,y] & 1<=k & k<=len s1 by A32; k in Seg len s1 & k=x by A33,XTUPLE_0:1; hence x in dom s1 by FINSEQ_1:def 3; end; then dom f c= dom s1 by TARSKI:def 3; then A34: dom s1 = dom f by A31,XBOOLE_0:def 10; then X = dom f by FINSEQ_1:def 3; then A35: rng f c= Seg l by A28,RELAT_1:113; then A36: dom s1 = dom(t*f) by A18,A34,RELAT_1:27; A37: now let x be set; assume A38: x in dom s1; then [x,f.x] in f by A31,FUNCT_1:def 2; then consider i be Element of NAT such that A39: [i,(i-1)*(len s2-1)+1] = [x,f.x] and A40: 1<=i and A41: i<=len s1; set k=(i-1)*(len s2-1)+1; (i-1)*(len s2-1)+1=f.x by A39,XTUPLE_0:1; then k in rng f by A31,A38,FUNCT_1:3; then k in Seg l by A35; then reconsider k as Element of NAT; A42: x in dom(t*f) by A18,A34,A35,A38,RELAT_1:27; per cases; suppose A43: i=len s1; (t*f).x = t.(f.x) by A42,FUNCT_1:12 .= t.k by A39,XTUPLE_0:1 .= (1).G by A1,A2,A43,Def35 .= s1.(len s1) by Def28; hence s1.x = (t*f).x by A39,A43,XTUPLE_0:1; end; suppose i<>len s1; then i1-1 by A2,XREAL_1:9; then A49: len s2-1>=0+1 by INT_1:7; 0+i<=1+i by XREAL_1:6; then 1<=i+1 by A40,XXREAL_0:2; then i+1 in Seg len s1 by A44; then A50: i+1 in dom s1 by FINSEQ_1:def 3; i in Seg len s1 by A40,A41; then i in dom s1 by FINSEQ_1:def 3; then A51: H1 is normal StableSubgroup of H2 by A50,Def28; (t*f).x = t.(f.x) by A42,FUNCT_1:12 .= t.k by A39,XTUPLE_0:1 .= H1"\/"(H2/\H3) by A1,A2,A40,A45,A49,Def35 .= H1"\/"H2 by A48,Th18 .= H2 by A51,Th36; hence s1.x = (t*f).x by A39,XTUPLE_0:1; end; end; now let r1,r2 be Element of REAL; assume r1 in X /\ dom f; then r1 in dom f by XBOOLE_0:def 4; then [r1,f.r1] in f by FUNCT_1:1; then consider k9 be Element of NAT such that A52: [k9,(k9-1)*(len s2-1)+1]=[r1,f.r1] and 1<=k9 and k9<=len s1; assume r2 in X /\ dom f; then r2 in dom f by XBOOLE_0:def 4; then [r2,f.r2] in f by FUNCT_1:1; then consider k99 be Element of NAT such that A53: [k99,(k99-1)*(len s2-1)+1]=[r2,f.r2] and 1<=k99 and k99<=len s1; A54: k99=r2 by A53,XTUPLE_0:1; assume A55: r1=1-1 by A60,A61,XREAL_1:9,XTUPLE_0:1; then y9 in NAT & not y in {0} by A19,INT_1:3,TARSKI:def 1; hence y in NAT \ {0} by XBOOLE_0:def 5; end; then f.:X c= NAT \ {0} by TARSKI:def 3; then the_schreier_series_of(s1,s2) * Sgm fX = the_schreier_series_of(s1, s2) * (f * Sgm X) by A30,A58,Lm38 .= (the_schreier_series_of(s1,s2) * f) * Sgm X by RELAT_1:36 .= s1 * Sgm X by A36,A37,FUNCT_1:2 .= s1 * idseq len s1 by FINSEQ_3:48 .= s1 * id Seg len s1 by FINSEQ_2:def 1 .= s1 * id dom s1 by FINSEQ_1:def 3; hence s1 = the_schreier_series_of(s1,s2) * Sgm fX by A15; end; hence thesis by Def29; end; theorem Th117: len s1>1 & len s2>1 implies the_schreier_series_of(s1,s2) is_equivalent_with the_schreier_series_of(s2,s1) proof assume that A1: len s1>1 and A2: len s2>1; set s21 = the_schreier_series_of(s2,s1); A3: len s1-1>1-1 & len s2-1>1-1 by A1,A2,XREAL_1:9; set s12 = the_schreier_series_of(s1,s2); A4: len s12 = (len s1-1)*(len s2-1) + 1 by A1,A2,Def35; A5: len s21 = (len s1-1)*(len s2-1) + 1 by A1,A2,Def35; then A6: s21 is not empty by A3; (len s1-1)*(len s2-1)>0*(len s2-1) by A3,XREAL_1:68; then A7: (len s1-1)*(len s2-1)+1>0+1 by XREAL_1:6; A8: now set p = {[(j-1)*(len s1-1)+i,(i-1)*(len s2-1)+j] where i,j is Element of NAT: 1<=i & i<=len s1-1 & 1<=j & j<=len s2-1}; now let x; assume x in p; then consider i,j be Element of NAT such that A9: [(j-1)*(len s1-1)+i,(i-1)*(len s2-1)+j]=x and 1<=i and i<=len s1-1 and 1<=j and j<=len s2-1; set z=(i-1)*(len s2-1)+j; set y=(j-1)*(len s1-1)+i; reconsider y,z as set; take y,z; thus x = [y,z] by A9; end; then reconsider p as Relation by RELAT_1:def 1; set X = dom the_series_of_quotients_of s12; set f1=the_series_of_quotients_of s12; set f2=the_series_of_quotients_of s21; now let x,y1,y2 be set; assume [x,y1] in p; then consider i1,j1 be Element of NAT such that A10: [(j1-1)*(len s1-1)+i1,(i1-1)*(len s2-1)+j1]=[x,y1] and A11: 1<=i1 & i1<=len s1-1 & 1<=j1 and j1<=len s2-1; A12: (j1-1)*(len s1-1)+i1=x by A10,XTUPLE_0:1; assume [x,y2] in p; then consider i2,j2 be Element of NAT such that A13: [(j2-1)*(len s1-1)+i2,(i2-1)*(len s2-1)+j2]=[x,y2] and A14: 1<=i2 & i2<=len s1-1 & 1<=j2 and j2<=len s2-1; A15: (j2-1)*(len s1-1)+i2=x by A13,XTUPLE_0:1; then j1=j2 by A1,A11,A14,A12,Lm46; hence y1=y2 by A10,A13,A12,A15,XTUPLE_0:1; end; then reconsider p as Function by FUNCT_1:def 1; A16: len s12>1 by A1,A2,A7,Def35; then A17: len f1 + 1= len s12 by Def33; A18: len s12 = (len s1-1)*(len s2-1)+1 by A1,A2,Def35; now set l9=(len s1-1)*(len s2-1); reconsider l9 as Element of NAT by A3,INT_1:3; let y; assume A19: y in X; then reconsider k=y as Element of NAT; A20: y in Seg len f1 by A19,FINSEQ_1:def 3; then A21: 1<=k by FINSEQ_1:1; A22: k<=(len s1-1)*(len s2-1) by A17,A18,A20,FINSEQ_1:1; 0+(len s1-1)*(len s2-1)<=1+(len s1-1)*(len s2-1) by XREAL_1:6; then k<=l9+1 by A22,XXREAL_0:2; then A23: k in Seg(l9+1) by A21; k <> l9+1 by A22,NAT_1:13; then consider i,j be Nat such that A24: k=(i-1)*(len s2-1)+j & 1<=i & i<=len s1-1 & 1<=j & j<=len s2-1 by A1,A2,A23,Lm45; reconsider j,i as Element of NAT by INT_1:3; set x=(j-1)*(len s1-1)+i; reconsider x as set; [x,y] in p by A24; hence y in rng p by XTUPLE_0:def 13; end; then A25: X c= rng p by TARSKI:def 3; A26: X = Seg len f1 by FINSEQ_1:def 3; now set l9=(len s1-1)*(len s2-1); reconsider l9 as Element of NAT by A3,INT_1:3; let x; assume A27: x in X; then reconsider k=x as Element of NAT; A28: k<=(len s1-1)*(len s2-1) by A17,A18,A26,A27,FINSEQ_1:1; 0+(len s1-1)*(len s2-1)<=1+(len s1-1)*(len s2-1) by XREAL_1:6; then A29: k<=l9+1 by A28,XXREAL_0:2; 1<=k by A26,A27,FINSEQ_1:1; then A30: k in Seg(l9+1) by A29; k <> l9+1 by A28,NAT_1:13; then consider j,i be Nat such that A31: k=(j-1)*(len s1-1)+i & 1<=j & j<=len s2-1 & 1<=i & i<=len s1-1 by A1,A2,A30,Lm45; reconsider j,i as Element of NAT by INT_1:3; set y=(i-1)*(len s2-1)+j; reconsider y as set; [x,y] in p by A31; hence x in dom p by XTUPLE_0:def 12; end; then A32: X c= dom p by TARSKI:def 3; now let y; set k=y; assume y in rng p; then consider x such that A33: [x,y] in p by XTUPLE_0:def 13; consider i,j be Element of NAT such that A34: [(j-1)*(len s1-1)+i,(i-1)*(len s2-1)+j]=[x,y] and A35: 1<=i & i<=len s1-1 & 1<=j & j<=len s2-1 by A33; A36: k=(i-1)*(len s2-1)+j by A34,XTUPLE_0:1; reconsider k as integer number by A34,XTUPLE_0:1; 1<=k by A2,A35,A36,Lm47; then reconsider k as Element of NAT by INT_1:3; 1<=k & k<=len f1 by A2,A17,A18,A35,A36,Lm47; hence y in X by A26; end; then rng p c= X by TARSKI:def 3; then A37: rng p = X by A25,XBOOLE_0:def 10; now let x; set k=x; assume x in dom p; then consider y such that A38: [x,y] in p by XTUPLE_0:def 12; consider i,j be Element of NAT such that A39: [(j-1)*(len s1-1)+i,(i-1)*(len s2-1)+j]=[x,y] and A40: 1<=i & i<=len s1-1 & 1<=j & j<=len s2-1 by A38; A41: k=(j-1)*(len s1-1)+i by A39,XTUPLE_0:1; reconsider k as integer number by A39,XTUPLE_0:1; 1<=k by A1,A40,A41,Lm47; then reconsider k as Element of NAT by INT_1:3; 1<=k & k<=len f1 by A1,A17,A18,A40,A41,Lm47; hence x in X by A26; end; then dom p c= X by TARSKI:def 3; then A42: dom p = X by A32,XBOOLE_0:def 10; then reconsider p as Function of X,X by A37,FUNCT_2:1; A43: p is onto by A37,FUNCT_2:def 3; now let x1,x2 be set; assume that A44: x1 in X and A45: x2 in X; assume A46: p.x1 = p.x2; [x1,p.x1] in p by A32,A44,FUNCT_1:def 2; then consider i1,j1 be Element of NAT such that A47: [(j1-1)*(len s1-1)+i1,(i1-1)*(len s2-1)+j1]=[x1,p.x1] and A48: 1<=i1 and i1<=len s1-1 and A49: 1<=j1 & j1<=len s2-1; [x2,p.x2] in p by A32,A45,FUNCT_1:def 2; then consider i2,j2 be Element of NAT such that A50: [(j2-1)*(len s1-1)+i2,(i2-1)*(len s2-1)+j2]=[x2,p.x2] and A51: 1<=i2 and i2<=len s1-1 and A52: 1<=j2 & j2<=len s2-1; A53: (i2-1)*(len s2-1)+j2=p.x2 by A50,XTUPLE_0:1; A54: (i1-1)*(len s2-1)+j1=p.x1 by A47,XTUPLE_0:1; then i1=i2 by A2,A46,A48,A49,A51,A52,A53,Lm46; hence x1 = x2 by A46,A47,A50,A54,A53,XTUPLE_0:1; end; then p is one-to-one by FUNCT_2:56; then reconsider p as Permutation of X by A43; take p; A55: len s21>1 by A1,A2,A7,Def35; then A56: len f2 + 1 = len s21 by Def33; now len s2+1>1+1 by A2,XREAL_1:6; then len s2>=2 by NAT_1:13; then A57: len s2-1>=2-1 by XREAL_1:9; set l=(len s1-1)*(len s2-1)+1; let H1,H2 be GroupWithOperators of O; let k1,k2 be Nat; assume that A58: k1 in dom f1 and A59: k2=p".k1; len s1+1>1+1 by A1,XREAL_1:6; then len s1>=2 by NAT_1:13; then len s1-1>=2-1 by XREAL_1:9; then reconsider l as Element of NAT by A57,INT_1:3; assume that A60: H1=f1.k1 and A61: H2=f2.k2; A62: len s12 = (len s1-1)*(len s2-1)+1 by A1,A2,Def35; 0+(len s1-1)*(len s2-1)<=1+(len s1-1)*(len s2-1) by XREAL_1:6; then A63: Seg len f1 c= Seg l by A17,A62,FINSEQ_1:5; A64: k1 in Seg len f1 by A58,FINSEQ_1:def 3; then k1 <= len f1 by FINSEQ_1:1; then k1<>(len s1-1)*(len s2-1)+1 by A17,A62,NAT_1:13; then consider i,j be Nat such that A65: k1=(i-1)*(len s2-1)+j and A66: 1<=i and A67: i<=len s1-1 and A68: 1<=j and A69: j<=len s2-1 by A1,A2,A64,A63,Lm45; reconsider H=s1.i,K=s2.j,H9=s1.(i+1),K9=s2.(j+1) as strict StableSubgroup of G by A66,A67,A68,A69,Th111; A70: p".k1 in rng(p") by A58,FUNCT_2:4; p.k2 = k1 by A25,A58,A59,FUNCT_1:35; then [k2,k1] in p by A42,A59,A70,FUNCT_1:1; then consider i9,j9 be Element of NAT such that A71: [k2,k1]=[(j9-1)*(len s1-1)+i9,(i9-1)*(len s2-1)+j9] and A72: 1<=i9 and i9<=len s1-1 and A73: 1<=j9 & j9<=len s2-1; set JK=K9"\/"(K/\H9); A74: (i-1)*(len s2-1)+j=(i9-1)*(len s2-1)+j9 by A65,A71,XTUPLE_0:1; then A75: i=i9 by A2,A66,A68,A69,A72,A73,Lm46; A76: now per cases; suppose A77: i=len s1-1; per cases; suppose A78: j<>len s2-1; set j9=j+1; A79: 0+j9<=1+j9 by XREAL_1:6; set i9=1; set H3 = s1.i9; H9 = (1).G by A77,Def28; then A80: JK = K9"\/"(1).G by Th21 .= K9 by Th33; set H2 = s2.j9; set H1 = s2.(j9+1); 1+1<=j+1 by A68,XREAL_1:6; then A81: 1<=j9 by XXREAL_0:2; j1-1 by A1,XREAL_1:9; then A85: len s1-1>=0+1 by INT_1:7; then reconsider H1,H2,H3 as strict StableSubgroup of G by A82,A81,Th111; A86: H3 = (Omega).G by Def28; now let x be set; H2 is Subgroup of G by Def7; then A87: the carrier of H2 c= the carrier of G by GROUP_2:def 5; assume x in the carrier of H2; hence x in the carrier of (Omega).G by A87; end; then the carrier of H2 c= the carrier of (Omega).G by TARSKI:def 3; then A88: the carrier of H2 = (the carrier of H2) /\ (the carrier of (Omega).G) by XBOOLE_1:28; k2+1=(j9-1)*(len s1-1)+i9 by A71,A74,A75,A77,XTUPLE_0:1; then s21.(k2+1)=H1"\/"(H2/\H3) by A1,A2,A82,A81,A85,Def35; then A89: s21.(k2+1) = H1"\/"H2 by A86,A88,Th18; 1<=j9+1 by A81,A79,XXREAL_0:2; then j9+1 in Seg len s2 by A83; then j9+1 in dom s2 by FINSEQ_1:def 3; then H1 is normal StableSubgroup of H2 by A84,Def28; hence s21.(k2+1)=JK by A89,A80,Th36; end; suppose A90: j=len s2-1; then A91: K9 = (1).G by Def28; H9 = (1).G by A77,Def28; then A92: JK = (1).G"\/"(1).G by A91,Th21 .= (1).G by Th33; k2 = (len s1-1)*(len s2-1) by A71,A74,A75,A77,A90,XTUPLE_0:1; hence s21.(k2+1)=JK by A1,A2,A92,Def35; end; end; suppose i<>len s1-1; then ilen s1-1; set j9=1; set H3 = s2.j9; set i9=i+1; A101: 0+i9<=1+i9 by XREAL_1:6; set H2 = s1.i9; set H1 = s1.(i9+1); 1+1<=i+1 by A66,XREAL_1:6; then A102: 1<=i9 by XXREAL_0:2; i1-1 by A2,XREAL_1:9; then A106: len s2-1>=0+1 by INT_1:7; then reconsider H1,H2,H3 as strict StableSubgroup of G by A103,A102,Th111; A107: H3 = (Omega).G by Def28; now let x be set; H2 is Subgroup of G by Def7; then A108: the carrier of H2 c= the carrier of G by GROUP_2:def 5; assume x in the carrier of H2; hence x in the carrier of (Omega).G by A108; end; then the carrier of H2 c= the carrier of (Omega).G by TARSKI:def 3; then A109: the carrier of H2 = (the carrier of H2) /\ (the carrier of (Omega).G) by XBOOLE_1:28; k1+1=(i9-1)*(len s2-1)+j9 by A65,A99; then s12.(k1+1)=H1"\/"(H2/\H3) by A1,A2,A103,A102,A106,Def35; then A110: s12.(k1+1) = H1"\/"H2 by A107,A109,Th18; 1<=i9+1 by A102,A101,XXREAL_0:2; then i9+1 in Seg len s1 by A104; then i9+1 in dom s1 by FINSEQ_1:def 3; then A111: H1 is normal StableSubgroup of H2 by A105,Def28; JH = H9"\/"(H/\(1).G) by A99,Def28 .= H9"\/"(1).G by Th21 .= H9 by Th33; hence s12.(k1+1)=JH by A110,A111,Th36; end; suppose A112: i=len s1-1; then A113: k1 = (len s1-1)*(len s2-1) by A65,A99; A114: K9 = (1).G by A99,Def28; H9 = (1).G by A112,Def28; then JH = (1).G"\/"(1).G by A114,Th21 .= (1).G by Th33; hence s12.(k1+1)=JH by A1,A2,A113,Def35; end; end; suppose j<>len s2-1; then j1 & len s2>1; set s29=the_schreier_series_of(s2,s1); set s19=the_schreier_series_of(s1,s2); take s19,s29; thus s19 is_finer_than s1 & s29 is_finer_than s2 by A1,Th116; thus thesis by A1,Th117; end; suppose A2: len s1<=1 or len s2<=1; per cases; suppose A3: len s1<=len s2; set s29=s2; set s19=s2; take s19,s29; thus s19 is_finer_than s1 & s29 is_finer_than s2 by A2,A3,Th114; thus thesis by Th113; end; suppose A4: len s1>len s2; set s29=s1; set s19=s1; take s19,s29; thus s19 is_finer_than s1 & s29 is_finer_than s2 by A2,A4,Th114; thus thesis by Th113; end; end; end; begin :: The Jordan-H\"older Theorem :: ALG I.4.7 Theorem 6 ::$N Jordan-H\"older Theorem theorem s1 is jordan_holder & s2 is jordan_holder implies s1 is_equivalent_with s2 proof assume A1: s1 is jordan_holder; assume A2: s2 is jordan_holder; per cases; suppose A3: s1 is empty; now now set x={}; take x; thus x c= dom s2 by XBOOLE_1:2; thus s1 = s2 * Sgm x by A3,FINSEQ_3:43; end; then A4: s2 is_finer_than s1 by Def29; assume A5: s2 is not empty; s2 is strictly_decreasing by A2,Def31; hence contradiction by A1,A3,A5,A4,Def31; end; hence thesis by A3,Th107; end; suppose A6: s1 is not empty; defpred P[Nat] means for s19,s29 st s19 is not empty & s29 is not empty & len s19=len s1+$1 & s19 is_finer_than s1 & s29 is_finer_than s2 & ex p being Permutation of dom the_series_of_quotients_of s19 st the_series_of_quotients_of s19,the_series_of_quotients_of s29 are_equivalent_under p,O holds ex p being Permutation of dom the_series_of_quotients_of s1 st the_series_of_quotients_of s1,the_series_of_quotients_of s2 are_equivalent_under p,O; A7: now assume A8: s2 is empty; now set x={}; take x; thus x c= dom s1 by XBOOLE_1:2; thus s2 = s1 * Sgm x by A8,FINSEQ_3:43; end; then A9: s1 is_finer_than s2 by Def29; s1 is strictly_decreasing by A1,Def31; hence contradiction by A2,A6,A8,A9,Def31; end; A10: for n st P[n] holds P[n+1] proof let n; assume A11: P[n]; now let s19,s29; assume that s19 is not empty and s29 is not empty; assume A12: len s19 = len s1 + n+1; set f1=the_series_of_quotients_of s19; assume A13: s19 is_finer_than s1; n+1+len s1>0+len s1 by XREAL_1:6; then consider i such that A14: i in dom f1 and A15: for H st H=f1.i holds H is trivial by A1,A12,A13,Th109; reconsider s199=Del(s19,i) as FinSequence of the_stable_subgroups_of G by FINSEQ_3:105; A16: i in dom s19 by A14,A15,Th103; A17: i+1 in dom s19 & s19.i=s19.(i+1) by A14,A15,Th103; then reconsider s199 as CompositionSeries of G by A16,Th94; A18: the_series_of_quotients_of s199= Del(f1,i) by A16,A17,Th104; set f2=the_series_of_quotients_of s29; assume A19: s29 is_finer_than s2; given p be Permutation of dom f1 such that A20: f1,f2 are_equivalent_under p,O; set H1=f1.i; A21: f1.i in rng f1 by A14,FUNCT_1:3; set j = p".i; reconsider j as Nat; set H2=f2.j; reconsider s299=Del(s29,j) as FinSequence of the_stable_subgroups_of G by FINSEQ_3:105; rng(p") c= dom f1; then A22: rng(p") c= Seg len f1 by FINSEQ_1:def 3; A23: len f1 = len f2 by A20,Def34; (p").i in rng(p") by A14,FUNCT_2:4; then (p").i in Seg len f1 by A22; then A24: j in dom f2 by A23,FINSEQ_1:def 3; then f2.j in rng f2 by FUNCT_1:3; then reconsider H1,H2 as strict GroupWithOperators of O by A21,Th102; A25: H1 is trivial by A15; H1,H2 are_isomorphic by A20,A14,Def34; then A26: for H st H=f2.j holds H is trivial by A25,Th58; then A27: j in dom s29 & j+1 in dom s29 by A24,Th103; A28: s29.j=s29.(j+1) by A24,A26,Th103; then reconsider s299 as CompositionSeries of G by A27,Th94; A29: s299 is_finer_than s2 & s299 is not empty by A2,A7,A19,A27,A28,Th97 ,Th99; A30: len s199 = len s1 + n by A12,A16,FINSEQ_3:109; the_series_of_quotients_of s299= Del(f2,j) by A27,A28,Th104; then A31: ex p be Permutation of dom the_series_of_quotients_of s199 st the_series_of_quotients_of s199,the_series_of_quotients_of s299 are_equivalent_under p,O by A20,A14,A18,Th106; s199 is_finer_than s1 & s199 is not empty by A1,A6,A13,A16,A17,Th97 ,Th99; hence thesis by A11,A30,A29,A31; end; hence thesis; end; A32: P[0] proof let s19,s29; assume A33: s19 is not empty & s29 is not empty; assume A34: len s19 = len s1+0 & s19 is_finer_than s1; assume A35: s29 is_finer_than s2; given p be Permutation of dom the_series_of_quotients_of s19 such that A36: the_series_of_quotients_of s19,the_series_of_quotients_of s29 are_equivalent_under p,O; A37: s19 is_equivalent_with s29 by A33,A36,Th108; s19=s1 by A34,Th96; then s29 is jordan_holder by A1,A37,Th115; then s29=s2 by A2,A35,Th98; then s1 is_equivalent_with s2 by A34,A37,Th96; hence thesis by A6,A7,Th108; end; A38: for n holds P[n] from NAT_1:sch 2(A32,A10); consider s19,s29 such that A39: s19 is_finer_than s1 and A40: s29 is_finer_than s2 and A41: s19 is_equivalent_with s29 by Th118; A42: s19 is not empty by A6,A39,Th97; A43: ex n st len s19 = len s1 + n by A39,Th95; A44: s29 is not empty by A7,A40,Th97; then ex p9 being Permutation of dom the_series_of_quotients_of s19 st the_series_of_quotients_of s19,the_series_of_quotients_of s29 are_equivalent_under p9,O by A41,A42,Th108; then ex p being Permutation of dom the_series_of_quotients_of s1 st the_series_of_quotients_of s1,the_series_of_quotients_of s2 are_equivalent_under p,O by A39,A40,A42,A44,A38,A43; hence thesis by A6,A7,Th108; end; end; begin :: Appendix theorem for P,R being Relation holds P = (rng P)|`R iff P~ = (R~)|(dom (P~)) by Lm36; theorem for X being set, P,R being Relation holds P*(R|X) = (X|`P)*R by Lm37; theorem for n being Nat, X being set, f being PartFunc of REAL, REAL st X c= Seg n & X c= dom f & f|X is increasing & f.:X c= NAT \ {0} holds Sgm(f.:X) = f * Sgm X by Lm38; theorem for y being set, i,n being Nat st y c= Seg(n+1) & i in Seg(n+1) & not i in y holds ex x st Sgm x = Sgm(Seg(n+1)\{i})" * Sgm y & x c= Seg n by Lm39; theorem for D being non empty set, f being FinSequence of D, p being Element of D, n being Nat st n in dom f holds f = Del(Ins(f,n,p),n+1) by Lm44; theorem for G,H being Group, F1 being FinSequence of the carrier of G, F2 being FinSequence of the carrier of H, I being FinSequence of INT, f being Homomorphism of G,H st (for k being Nat st k in dom F1 holds F2.k = f.(F1.k)) & len F1 = len I & len F2 = len I holds f.(Product(F1 |^ I)) = Product(F2 |^ I) by Lm24;