:: Cages, external approximation of Jordan's curve :: by Czes{\l}aw Byli\'nski and Mariusz \.Zynel :: :: Received June 22, 1999 :: Copyright (c) 1999-2012 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, SUBSET_1, XBOOLE_0, FINSEQ_1, XXREAL_0, ARYTM_1, ARYTM_3, PRE_TOPC, RELAT_2, CONNSP_1, TARSKI, RELAT_1, FINSEQ_5, PARTFUN1, GOBOARD1, EUCLID, REAL_1, MATRIX_1, COMPLEX1, GOBRD13, FUNCT_1, TOPREAL1, RFINSEQ, RLTOPSP1, GOBOARD5, TOPS_1, TREES_1, SPPOL_1, MCART_1, CARD_1, GOBOARD9, RCOMP_1, NAT_1, JORDAN8, PSCOMP_1, NEWTON, SPRECT_2, ORDINAL4, STRUCT_0, PCOMPS_1, XREAL_0, ORDINAL1, METRIC_1, JORDAN9, CONVEX1; notations TARSKI, XBOOLE_0, SUBSET_1, GOBOARD5, ORDINAL1, CARD_1, NUMBERS, XCMPLX_0, XREAL_0, REAL_1, NAT_1, NAT_D, RELAT_1, FUNCT_1, PARTFUN1, FUNCT_2, COMPLEX1, FINSEQ_1, FINSEQ_2, FINSEQ_4, FINSEQ_5, NEWTON, RFINSEQ, STRUCT_0, XXREAL_0, MATRIX_1, METRIC_1, PRE_TOPC, TOPS_1, COMPTS_1, CONNSP_1, PCOMPS_1, RLTOPSP1, EUCLID, TOPREAL1, GOBOARD1, SPPOL_1, PSCOMP_1, SPRECT_2, GOBOARD9, JORDAN8, GOBRD13; constructors REAL_1, FINSEQ_4, NEWTON, RFINSEQ, NAT_D, TOPS_1, CONNSP_1, COMPTS_1, SPPOL_1, PSCOMP_1, GOBOARD9, SPRECT_2, JORDAN8, GOBRD13, RELSET_1, FUNCSDOM, PCOMPS_1, CONVEX1; registrations RELAT_1, FUNCT_1, ORDINAL1, XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, STRUCT_0, EUCLID, SPPOL_2, PSCOMP_1, GOBOARD9, SPRECT_1, SPRECT_2, JORDAN8, NEWTON, FINSET_1, SPPOL_1, JORDAN1, MATRIX_1; requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM; definitions TARSKI, GOBOARD5, GOBRD13, XBOOLE_0, PSCOMP_1, SEQM_3; theorems NAT_1, FINSEQ_1, GOBOARD1, FINSEQ_4, EUCLID, FINSEQ_3, SPPOL_2, TARSKI, JORDAN3, PSCOMP_1, FINSEQ_5, FINSEQ_6, GOBOARD7, TOPREAL1, JORDAN5B, GOBOARD5, SPRECT_2, SPPOL_1, ABSVALUE, FUNCT_1, FUNCT_2, GOBOARD9, FINSEQ_2, UNIFORM1, SUBSET_1, GOBRD11, SPRECT_3, CARD_1, RFINSEQ, GOBOARD6, TOPREAL3, TOPMETR, TOPS_1, JORDAN8, GOBRD13, SPRECT_4, CONNSP_1, PARTFUN2, RELSET_1, SPRECT_1, XBOOLE_0, XBOOLE_1, XREAL_0, XCMPLX_1, XREAL_1, NEWTON, XXREAL_0, ORDINAL1, PARTFUN1, MATRIX_1, NAT_D, RLTOPSP1, SEQ_4; schemes NAT_1, RECDEF_1; begin :: Generalities reserve i,j,k,n for Element of NAT, D for non empty set, f, g for FinSequence of D; Lm1: for n st 1 <= n holds n-'1+2 = n+1 proof let n; assume 1 <= n; hence n-'1+2 = n+2-'1 by NAT_D:38 .= n+1+1 - 1 by NAT_D:37 .= n+1; end; theorem Th1: for T being non empty TopSpace for B,C1,C2,D being Subset of T st B is connected & C1 is_a_component_of D & C2 is_a_component_of D & B meets C1 & B meets C2 & B c= D holds C1 = C2 proof let T be non empty TopSpace; let B,C1,C2,D be Subset of T; assume that A1: B is connected and A2: C1 is_a_component_of D & C2 is_a_component_of D and A3: B meets C1 and A4: B meets C2 & B c= D; A5: B <> {} by A3,XBOOLE_1:65; B c= C1 & B c= C2 by A1,A2,A3,A4,GOBOARD9:4; hence thesis by A2,A5,GOBOARD9:1,XBOOLE_1:68; end; theorem Th2: (for n holds f|n = g|n) implies f = g proof assume A1: for n holds f|n = g|n; A2: now assume A3: len f <> len g; per cases by A3,XXREAL_0:1; suppose A4: len f < len g; A5: g|len g = g by FINSEQ_1:58; f|len g = f by A4,FINSEQ_1:58; hence contradiction by A1,A4,A5; end; suppose A6: len g < len f; then f|len f = f & g|len f = g by FINSEQ_1:58; hence contradiction by A1,A6; end; end; f|len f = f & g|len g = g by FINSEQ_1:58; hence thesis by A1,A2; end; theorem Th3: n in dom f implies ex k st k in dom Rev f & n+k = len f+1 & f/.n = (Rev f)/.k proof assume A1: n in dom f; take k = len f+1-'n; 1 <= n by A1,FINSEQ_3:25; then k+1 <= k+n by XREAL_1:6; then A2: k+1-'1 <= k+n -'1 by NAT_D:42; A3: n <= len f by A1,FINSEQ_3:25; then n+1 <= len f+1 by XREAL_1:6; then A4: 1 <= k by NAT_D:55; n <= len f+1 by A3,XREAL_1:145; then A5: k+n = len f+1 by XREAL_1:235; then k+n-'1 = len f by NAT_D:34; then k <= len f by A2,NAT_D:34; then k in dom f by A4,FINSEQ_3:25; hence thesis by A1,A5,FINSEQ_5:57,66; end; theorem Th4: n in dom Rev f implies ex k st k in dom f & n+k = len f+1 & (Rev f)/.n = f/.k proof assume n in dom Rev f; then n in dom f by FINSEQ_5:57; then consider k such that A1: k in dom Rev f & n+k = len f+1 and f/.n = (Rev f)/.k by Th3; A2: dom f = dom Rev f by FINSEQ_5:57; then (Rev f)/.n = f/.k by A1,FINSEQ_5:66; hence thesis by A1,A2; end; begin :: Go-board preliminaries reserve G for Go-board, f, g for FinSequence of TOP-REAL 2, p for Point of TOP-REAL 2, r, s for Real, x for set; theorem Th5: for D being non empty set for G being Matrix of D for f being FinSequence of D holds f is_sequence_on G iff Rev f is_sequence_on G proof let D be non empty set; let G be Matrix of D; let f be FinSequence of D; hereby assume A1: f is_sequence_on G; A2: for n st n in dom Rev f & n+1 in dom Rev f holds for m,k,i,j being Element of NAT st [m,k] in Indices G & [i,j] in Indices G & (Rev f)/.n = G*(m,k ) & (Rev f)/.(n+1) = G*(i,j) holds abs(m-i)+abs(k-j) = 1 proof let n such that A3: n in dom Rev f and A4: n+1 in dom Rev f; consider l being Element of NAT such that A5: l in dom f and A6: n+l = len f+1 and A7: (Rev f)/.n = f/.l by A3,Th4; let m,k,i,j be Element of NAT such that A8: [m,k] in Indices G & [i,j] in Indices G & (Rev f)/.n = G*(m,k) & (Rev f)/.(n+1) = G*(i,j); A9: abs(i-m) = abs(m-i) & abs(j-k) = abs(k-j) by UNIFORM1:11; consider l9 being Element of NAT such that A10: l9 in dom f and A11: n+1+l9 = len f+1 and A12: (Rev f)/.(n+1) = f/.l9 by A4,Th4; n+(1+l9) = n+l by A6,A11; hence thesis by A1,A8,A5,A7,A10,A12,A9,GOBOARD1:def 9; end; for n st n in dom Rev f ex i,j st [i,j] in Indices G & (Rev f)/.n = G* (i,j) proof let n; assume n in dom Rev f; then consider k such that A13: k in dom f and n+k = len f+1 and A14: (Rev f)/.n = f/.k by Th4; consider i,j such that A15: [i,j] in Indices G & f/.k = G*(i,j) by A1,A13,GOBOARD1:def 9; take i,j; thus thesis by A14,A15; end; hence Rev f is_sequence_on G by A2,GOBOARD1:def 9; end; assume A16: Rev f is_sequence_on G; A17: for n st n in dom f & n+1 in dom f holds for m,k,i,j being Element of NAT st [m,k] in Indices G & [i,j] in Indices G & f/.n = G*(m,k) & f/.(n+1) = G* (i,j) holds abs(m-i)+abs(k-j) = 1 proof let n such that A18: n in dom f and A19: n+1 in dom f; consider l being Element of NAT such that A20: l in dom Rev f and A21: n+l = len f+1 and A22: f/.n = (Rev f)/.l by A18,Th3; let m,k,i,j be Element of NAT such that A23: [m,k] in Indices G & [i,j] in Indices G & f/.n = G*(m,k) & f/.(n+ 1) = G*(i,j); A24: abs(i-m) = abs(m-i) & abs(j-k) = abs(k-j) by UNIFORM1:11; consider l9 being Element of NAT such that A25: l9 in dom Rev f and A26: n+1+l9 = len f+1 and A27: f/.(n+1) = (Rev f)/.l9 by A19,Th3; n+(1+l9) = n+l by A21,A26; hence thesis by A16,A23,A20,A22,A25,A27,A24,GOBOARD1:def 9; end; for n st n in dom f ex i,j st [i,j] in Indices G & f/.n = G*(i,j) proof let n; assume n in dom f; then consider k such that A28: k in dom Rev f and n+k = len f+1 and A29: f/.n = (Rev f)/.k by Th3; consider i,j such that A30: [i,j] in Indices G & (Rev f)/.k = G*(i,j) by A16,A28,GOBOARD1:def 9; take i,j; thus thesis by A29,A30; end; hence thesis by A17,GOBOARD1:def 9; end; theorem Th6: for G being Matrix of TOP-REAL 2 for f being FinSequence of TOP-REAL 2 holds f is_sequence_on G & 1 <= k & k <= len f implies f/.k in Values G proof let G be Matrix of TOP-REAL 2; let f be FinSequence of TOP-REAL 2; assume that A1: f is_sequence_on G and A2: 1 <= k & k <= len f; A3: k in dom f by A2,FINSEQ_3:25; then f/.k = f.k by PARTFUN1:def 6; then A4: f/.k in rng f by A3,FUNCT_1:def 3; rng f c= Values G by A1,GOBRD13:8; hence thesis by A4; end; Lm2: f is_sequence_on G & 1 <= k & k <= len f implies ex i,j being Element of NAT st [i,j] in Indices G & f/.k = G*(i,j) proof assume that A1: f is_sequence_on G and A2: 1 <= k & k <= len f; k in dom f by A2,FINSEQ_3:25; then consider i,j such that A3: [i,j] in Indices G & f/.k = G*(i,j) by A1,GOBOARD1:def 9; take i,j; thus thesis by A3; end; theorem Th7: n <= len f & x in L~(f/^n) implies ex i being Element of NAT st n +1 <= i & i+1 <= len f & x in LSeg(f,i) proof assume that A1: n <= len f and A2: x in L~(f/^n); consider j such that A3: 1 <= j and A4: j+1 <= len(f/^n) and A5: x in LSeg(f/^n,j) by A2,SPPOL_2:13; j+1 <= len f - n by A1,A4,RFINSEQ:def 1; then A6: n+(j+1) <= len f by XREAL_1:19; take n+j; j+1 <= len f - n by A1,A4,RFINSEQ:def 1; hence thesis by A3,A5,A6,SPPOL_2:5,XREAL_1:6; end; theorem Th8: f is_sequence_on G & 1 <= k & k+1 <= len f implies f/.k in left_cell(f,k,G) & f/.k in right_cell(f,k,G) proof assume that A1: f is_sequence_on G and A2: 1 <= k & k+1 <= len f; set p = f/.k; LSeg(f,k) = LSeg(f/.k, f/.(k+1)) by A2,TOPREAL1:def 3; then p in LSeg(f,k) by RLTOPSP1:68; then p in left_cell(f,k,G) /\ right_cell(f,k,G) by A1,A2,GOBRD13:29; hence thesis by XBOOLE_0:def 4; end; theorem Th9: f is_sequence_on G & 1 <= k & k+1 <= len f implies Int left_cell (f,k,G) <> {} & Int right_cell(f,k,G) <> {} proof assume A1: f is_sequence_on G & 1 <= k & k+1 <= len f; then consider i1,j1,i2,j2 being Element of NAT such that A2: [i1,j1] in Indices G and A3: f/.k = G*(i1,j1) and A4: [i2,j2] in Indices G and A5: f/.(k+1) = G*(i2,j2) and A6: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by JORDAN8:3; A7: i2 <= len G by A4,MATRIX_1:38; A8: i1 <= len G by A2,MATRIX_1:38; then A9: i1-'1 <= len G by NAT_D:44; A10: j2 <= width G by A4,MATRIX_1:38; then A11: j2-'1 <= width G by NAT_D:44; A12: j1+1 > j1 & j2+1 > j2 by NAT_1:13; A13: j1 <= width G by A2,MATRIX_1:38; then A14: j1-'1 <= width G by NAT_D:44; A15: i1+1 > i1 & i2+1 > i2 by NAT_1:13; per cases by A6; suppose i1 = i2 & j1+1 = j2; then right_cell(f,k,G) = cell(G,i1,j1) & left_cell(f,k,G) = cell(G,i1-'1, j1) by A1,A2,A3,A4,A5,A12,GOBRD13:def 2,def 3; hence thesis by A8,A13,A9,GOBOARD9:14; end; suppose i1+1 = i2 & j1 = j2; then right_cell(f,k,G) = cell(G,i1,j1-'1) & left_cell(f,k,G) = cell(G,i1, j1) by A1,A2,A3,A4,A5,A15,GOBRD13:def 2,def 3; hence thesis by A8,A13,A14,GOBOARD9:14; end; suppose i1 = i2+1 & j1 = j2; then right_cell(f,k,G) = cell(G,i2,j2) & left_cell(f,k,G) = cell(G,i2,j2-' 1) by A1,A2,A3,A4,A5,A15,GOBRD13:def 2,def 3; hence thesis by A7,A10,A11,GOBOARD9:14; end; suppose i1 = i2 & j1 = j2+1; then right_cell(f,k,G) = cell(G,i1-'1,j2) & left_cell(f,k,G) = cell(G,i1, j2) by A1,A2,A3,A4,A5,A12,GOBRD13:def 2,def 3; hence thesis by A8,A10,A9,GOBOARD9:14; end; end; theorem Th10: f is_sequence_on G & 1 <= k & k+1 <= len f implies Int left_cell (f,k,G) is convex & Int right_cell(f,k,G) is convex proof assume A1: f is_sequence_on G & 1 <= k & k+1 <= len f; then consider i1,j1,i2,j2 being Element of NAT such that A2: [i1,j1] in Indices G and A3: f/.k = G*(i1,j1) and A4: [i2,j2] in Indices G and A5: f/.(k+1) = G*(i2,j2) and A6: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by JORDAN8:3; A7: i2 <= len G by A4,MATRIX_1:38; A8: i1 <= len G by A2,MATRIX_1:38; then A9: i1-'1 <= len G by NAT_D:44; A10: j2 <= width G by A4,MATRIX_1:38; then A11: j2-'1 <= width G by NAT_D:44; A12: j1+1 > j1 & j2+1 > j2 by NAT_1:13; A13: j1 <= width G by A2,MATRIX_1:38; then A14: j1-'1 <= width G by NAT_D:44; A15: i1+1 > i1 & i2+1 > i2 by NAT_1:13; per cases by A6; suppose i1 = i2 & j1+1 = j2; then right_cell(f,k,G) = cell(G,i1,j1) & left_cell(f,k,G) = cell(G,i1-'1, j1) by A1,A2,A3,A4,A5,A12,GOBRD13:def 2,def 3; hence thesis by A8,A13,A9,GOBOARD9:17; end; suppose i1+1 = i2 & j1 = j2; then right_cell(f,k,G) = cell(G,i1,j1-'1) & left_cell(f,k,G) = cell(G,i1, j1) by A1,A2,A3,A4,A5,A15,GOBRD13:def 2,def 3; hence thesis by A8,A13,A14,GOBOARD9:17; end; suppose i1 = i2+1 & j1 = j2; then right_cell(f,k,G) = cell(G,i2,j2) & left_cell(f,k,G) = cell(G,i2,j2-' 1) by A1,A2,A3,A4,A5,A15,GOBRD13:def 2,def 3; hence thesis by A7,A10,A11,GOBOARD9:17; end; suppose i1 = i2 & j1 = j2+1; then right_cell(f,k,G) = cell(G,i1-'1,j2) & left_cell(f,k,G) = cell(G,i1, j2) by A1,A2,A3,A4,A5,A12,GOBRD13:def 2,def 3; hence thesis by A8,A10,A9,GOBOARD9:17; end; end; theorem Th11: f is_sequence_on G & 1 <= k & k+1 <= len f implies Cl Int left_cell(f,k,G) = left_cell(f,k,G) & Cl Int right_cell(f,k,G) = right_cell(f,k ,G) proof assume A1: f is_sequence_on G & 1 <= k & k+1 <= len f; then consider i1,j1,i2,j2 being Element of NAT such that A2: [i1,j1] in Indices G and A3: f/.k = G*(i1,j1) and A4: [i2,j2] in Indices G and A5: f/.(k+1) = G*(i2,j2) and A6: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by JORDAN8:3; A7: i2 <= len G by A4,MATRIX_1:38; A8: i1 <= len G by A2,MATRIX_1:38; then A9: i1-'1 <= len G by NAT_D:44; A10: j2 <= width G by A4,MATRIX_1:38; then A11: j2-'1 <= width G by NAT_D:44; A12: j1+1 > j1 & j2+1 > j2 by NAT_1:13; A13: j1 <= width G by A2,MATRIX_1:38; then A14: j1-'1 <= width G by NAT_D:44; A15: i1+1 > i1 & i2+1 > i2 by NAT_1:13; per cases by A6; suppose i1 = i2 & j1+1 = j2; then right_cell(f,k,G) = cell(G,i1,j1) & left_cell(f,k,G) = cell(G,i1-'1, j1) by A1,A2,A3,A4,A5,A12,GOBRD13:def 2,def 3; hence thesis by A8,A13,A9,GOBRD11:35; end; suppose i1+1 = i2 & j1 = j2; then right_cell(f,k,G) = cell(G,i1,j1-'1) & left_cell(f,k,G) = cell(G,i1, j1) by A1,A2,A3,A4,A5,A15,GOBRD13:def 2,def 3; hence thesis by A8,A13,A14,GOBRD11:35; end; suppose i1 = i2+1 & j1 = j2; then right_cell(f,k,G) = cell(G,i2,j2) & left_cell(f,k,G) = cell(G,i2,j2-' 1) by A1,A2,A3,A4,A5,A15,GOBRD13:def 2,def 3; hence thesis by A7,A10,A11,GOBRD11:35; end; suppose i1 = i2 & j1 = j2+1; then right_cell(f,k,G) = cell(G,i1-'1,j2) & left_cell(f,k,G) = cell(G,i1, j2) by A1,A2,A3,A4,A5,A12,GOBRD13:def 2,def 3; hence thesis by A8,A10,A9,GOBRD11:35; end; end; theorem Th12: f is_sequence_on G & LSeg(f,k) is horizontal implies ex j st 1 <= j & j <= width G & for p st p in LSeg(f,k) holds p`2 = G*(1,j)`2 proof assume that A1: f is_sequence_on G and A2: LSeg(f,k) is horizontal; per cases; suppose A3: 1 <= k & k+1 <= len f; k <= k+1 by NAT_1:11; then k <= len f by A3,XXREAL_0:2; then consider i,j such that A4: [i,j] in Indices G and A5: f/.k = G*(i,j) by A1,A3,Lm2; take j; thus A6: 1 <= j & j <= width G by A4,MATRIX_1:38; A7: f/.k in LSeg(f,k) by A3,TOPREAL1:21; let p; A8: 1 <= i & i <= len G by A4,MATRIX_1:38; assume p in LSeg(f,k); hence p`2 = (f/.k)`2 by A2,A7,SPPOL_1:def 2 .= G*(1,j)`2 by A5,A6,A8,GOBOARD5:1; end; suppose A9: not(1 <= k & k+1 <= len f); take 1; width G <> 0 by GOBOARD1:def 3; hence 1 <= 1 & 1 <= width G by NAT_1:14; thus thesis by A9,TOPREAL1:def 3; end; end; theorem Th13: f is_sequence_on G & LSeg(f,k) is vertical implies ex i st 1 <= i & i <= len G & for p st p in LSeg(f,k) holds p`1 = G*(i,1)`1 proof assume that A1: f is_sequence_on G and A2: LSeg(f,k) is vertical; per cases; suppose A3: 1 <= k & k+1 <= len f; k <= k+1 by NAT_1:11; then k <= len f by A3,XXREAL_0:2; then consider i,j such that A4: [i,j] in Indices G and A5: f/.k = G*(i,j) by A1,A3,Lm2; take i; thus A6: 1 <= i & i <= len G by A4,MATRIX_1:38; A7: f/.k in LSeg(f,k) by A3,TOPREAL1:21; let p; A8: 1 <= j & j <= width G by A4,MATRIX_1:38; assume p in LSeg(f,k); hence p`1 = (f/.k)`1 by A2,A7,SPPOL_1:def 3 .= G*(i,1)`1 by A5,A6,A8,GOBOARD5:2; end; suppose A9: not(1 <= k & k+1 <= len f); take 1; 0 <> len G by GOBOARD1:def 3; hence 1 <= 1 & 1 <= len G by NAT_1:14; thus thesis by A9,TOPREAL1:def 3; end; end; theorem Th14: f is_sequence_on G & f is special & i <= len G & j <= width G implies Int cell(G,i,j) misses L~f proof assume that A1: f is_sequence_on G and A2: f is special and A3: i <= len G and A4: j <= width G; A5: Int cell(G,i,j) = Int v_strip(G,i) /\ Int h_strip(G,j) by TOPS_1:17; assume Int cell(G,i,j) meets L~f; then consider x being set such that A6: x in Int cell(G,i,j) and A7: x in L~f by XBOOLE_0:3; L~f = union { LSeg(f,k) : 1 <= k & k+1 <= len f } by TOPREAL1:def 4; then consider X being set such that A8: x in X and A9: X in { LSeg(f,k) : 1 <= k & k+1 <= len f } by A7,TARSKI:def 4; consider k such that A10: X = LSeg(f,k) and 1 <= k and k+1 <= len f by A9; reconsider p = x as Point of TOP-REAL 2 by A8,A10; per cases by A2,SPPOL_1:19; suppose LSeg(f,k) is horizontal; then consider j0 being Element of NAT such that A11: 1 <= j0 and A12: j0 <= width G and A13: for p being Point of TOP-REAL 2 st p in LSeg(f,k) holds p`2 = G*( 1,j0)`2 by A1,Th12; now A14: j0 > j implies j0 >= j+1 by NAT_1:13; assume A15: p in Int h_strip(G,j); per cases by A14,XXREAL_0:1; suppose A16: j0 < j; 0 <> len G by GOBOARD1:def 3; then 1 <= len G by NAT_1:14; then A17: G*(1,j)`2 > G*(1,j0)`2 by A4,A11,A16,GOBOARD5:4; j >= 1 by A11,A16,XXREAL_0:2; then p`2 > G*(1,j)`2 by A4,A15,GOBOARD6:27; hence contradiction by A8,A10,A13,A17; end; suppose j0 = j; then p`2 > G*(1,j0)`2 by A11,A12,A15,GOBOARD6:27; hence contradiction by A8,A10,A13; end; suppose A18: j0 > j+1; then j+1 <= width G by A12,XXREAL_0:2; then j < width G by NAT_1:13; then A19: p`2 < G*(1,j+1)`2 by A15,GOBOARD6:28; 0 <> len G by GOBOARD1:def 3; then A20: 1 <= len G by NAT_1:14; j+1 >= 1 by NAT_1:14; then G*(1,j+1)`2 < G*(1,j0)`2 by A12,A18,A20,GOBOARD5:4; hence contradiction by A8,A10,A13,A19; end; suppose A21: j0 = j+1; then j < width G by A12,NAT_1:13; then p`2 < G*(1,j0)`2 by A15,A21,GOBOARD6:28; hence contradiction by A8,A10,A13; end; end; hence contradiction by A6,A5,XBOOLE_0:def 4; end; suppose LSeg(f,k) is vertical; then consider i0 being Element of NAT such that A22: 1 <= i0 and A23: i0 <= len G and A24: for p being Point of TOP-REAL 2 st p in LSeg(f,k) holds p`1 = G*( i0,1)`1 by A1,Th13; now A25: i0 > i implies i0 >= i+1 by NAT_1:13; assume A26: p in Int v_strip(G,i); per cases by A25,XXREAL_0:1; suppose A27: i0 < i; 0 <> width G by GOBOARD1:def 3; then 1 <= width G by NAT_1:14; then A28: G*(i,1)`1 > G*(i0,1)`1 by A3,A22,A27,GOBOARD5:3; i >= 1 by A22,A27,XXREAL_0:2; then p`1 > G*(i,1)`1 by A3,A26,GOBOARD6:29; hence contradiction by A8,A10,A24,A28; end; suppose i0 = i; then p`1 > G*(i0,1)`1 by A22,A23,A26,GOBOARD6:29; hence contradiction by A8,A10,A24; end; suppose A29: i0 > i+1; then i+1 <= len G by A23,XXREAL_0:2; then i < len G by NAT_1:13; then A30: p`1 < G*(i+1,1)`1 by A26,GOBOARD6:30; 0 <> width G by GOBOARD1:def 3; then A31: 1 <= width G by NAT_1:14; i+1 >= 1 by NAT_1:14; then G*(i+1,1)`1 < G*(i0,1)`1 by A23,A29,A31,GOBOARD5:3; hence contradiction by A8,A10,A24,A30; end; suppose A32: i0 = i+1; then i < len G by A23,NAT_1:13; then p`1 < G*(i0,1)`1 by A26,A32,GOBOARD6:30; hence contradiction by A8,A10,A24; end; end; hence contradiction by A6,A5,XBOOLE_0:def 4; end; end; theorem Th15: f is_sequence_on G & f is special & 1 <= k & k+1 <= len f implies Int left_cell(f,k,G) misses L~f & Int right_cell(f,k,G) misses L~f proof assume that A1: f is_sequence_on G and A2: f is special and A3: 1 <= k & k+1 <= len f; consider i1,j1,i2,j2 being Element of NAT such that A4: [i1,j1] in Indices G and A5: f/.k = G*(i1,j1) and A6: [i2,j2] in Indices G and A7: f/.(k+1) = G*(i2,j2) and A8: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A1,A3,JORDAN8:3; A9: i2 <= len G by A6,MATRIX_1:38; A10: i1 <= len G by A4,MATRIX_1:38; then A11: i1-'1 <= len G by NAT_D:44; A12: j2 <= width G by A6,MATRIX_1:38; then A13: j2-'1 <= width G by NAT_D:44; A14: j1+1 > j1 & j2+1 > j2 by NAT_1:13; A15: j1 <= width G by A4,MATRIX_1:38; then A16: j1-'1 <= width G by NAT_D:44; A17: i1+1 > i1 & i2+1 > i2 by NAT_1:13; per cases by A8; suppose i1 = i2 & j1+1 = j2; then right_cell(f,k,G) = cell(G,i1,j1) & left_cell(f,k,G) = cell(G,i1-'1, j1) by A1,A3,A4,A5,A6,A7,A14,GOBRD13:def 2,def 3; hence thesis by A1,A2,A10,A15,A11,Th14; end; suppose i1+1 = i2 & j1 = j2; then right_cell(f,k,G) = cell(G,i1,j1-'1) & left_cell(f,k,G) = cell(G,i1, j1) by A1,A3,A4,A5,A6,A7,A17,GOBRD13:def 2,def 3; hence thesis by A1,A2,A10,A15,A16,Th14; end; suppose i1 = i2+1 & j1 = j2; then right_cell(f,k,G) = cell(G,i2,j2) & left_cell(f,k,G) = cell(G,i2,j2-' 1) by A1,A3,A4,A5,A6,A7,A17,GOBRD13:def 2,def 3; hence thesis by A1,A2,A9,A12,A13,Th14; end; suppose i1 = i2 & j1 = j2+1; then right_cell(f,k,G) = cell(G,i1-'1,j2) & left_cell(f,k,G) = cell(G,i1, j2) by A1,A3,A4,A5,A6,A7,A14,GOBRD13:def 2,def 3; hence thesis by A1,A2,A10,A12,A11,Th14; end; end; theorem Th16: 1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G implies G*(i,j) `1 = G*(i,j+1)`1 & G*(i,j)`2 = G*(i+1,j)`2 & G*(i+1,j+1)`1 = G*(i+1,j)`1 & G*(i +1,j+1)`2 = G*(i,j+1)`2 proof assume that A1: 1 <= i and A2: i+1 <= len G and A3: 1 <= j and A4: j+1 <= width G; A5: j < width G by A4,NAT_1:13; A6: 1 <= j+1 by NAT_1:11; A7: i < len G by A2,NAT_1:13; hence G*(i,j)`1 = G*(i,1)`1 by A1,A3,A5,GOBOARD5:2 .= G*(i,j+1)`1 by A1,A4,A7,A6,GOBOARD5:2; A8: 1 <= i+1 by NAT_1:11; thus G*(i,j)`2 = G*(1,j)`2 by A1,A3,A7,A5,GOBOARD5:1 .= G*(i+1,j)`2 by A2,A3,A5,A8,GOBOARD5:1; thus G*(i+1,j+1)`1 = G*(i+1,1)`1 by A2,A4,A8,A6,GOBOARD5:2 .= G*(i+1,j)`1 by A2,A3,A5,A8,GOBOARD5:2; thus G*(i+1,j+1)`2 = G*(1,j+1)`2 by A2,A4,A8,A6,GOBOARD5:1 .= G*(i,j+1)`2 by A1,A4,A7,A6,GOBOARD5:1; end; theorem Th17: for i,j being Element of NAT st 1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G holds p in cell(G,i,j) iff G*(i,j)`1 <= p`1 & p`1 <= G*(i+1,j)`1 & G*(i,j)`2 <= p`2 & p`2 <= G*(i,j+1)`2 proof let i,j be Element of NAT such that A1: 1 <= i and A2: i+1 <= len G and A3: 1 <= j and A4: j+1 <= width G; A5: i < len G & j < width G by A2,A4,NAT_1:13; then A6: h_strip(G,j) = { |[r,s]| : G*(i,j)`2 <= s & s <= G*(i,j+1)`2 } by A1,A3, GOBOARD5:5; A7: v_strip(G,i) = { |[r,s]| : G*(i,j)`1 <= r & r <= G*(i+1,j)`1 } by A1,A3,A5, GOBOARD5:8; hereby assume A8: p in cell(G,i,j); then p in v_strip(G,i) by XBOOLE_0:def 4; then ex r,s st |[r,s]| = p & G*(i,j)`1 <= r & r <= G*(i+1,j)`1 by A7; hence G*(i,j)`1 <= p`1 & p`1 <= G*(i+1,j)`1 by EUCLID:52; p in h_strip(G,j) by A8,XBOOLE_0:def 4; then ex r,s st |[r,s]| = p & G*(i,j)`2 <= s & s <= G*(i,j+1)`2 by A6; hence G*(i,j)`2 <= p`2 & p`2 <= G*(i,j+1)`2 by EUCLID:52; end; assume that A9: G*(i,j)`1 <= p`1 & p`1 <= G*(i+1,j)`1 and A10: G*(i,j)`2 <= p`2 & p`2 <= G*(i,j+1)`2; A11: p = |[p`1,p`2]| by EUCLID:53; then A12: p in h_strip(G,j) by A6,A10; p in v_strip(G,i) by A7,A9,A11; hence thesis by A12,XBOOLE_0:def 4; end; theorem 1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G implies cell(G,i,j) = { |[r,s]| : G*(i,j)`1 <= r & r <= G*(i+1,j)`1 & G*(i,j)`2 <= s & s <= G*(i,j+1) `2 } proof set A = { |[r,s]| : G*(i,j)`1 <= r & r <= G*(i+1,j)`1 & G*(i,j)`2 <= s & s <= G*(i,j+1)`2 }; assume A1: 1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G; now let p be set; assume A2: p in cell(G,i,j); then reconsider q=p as Point of TOP-REAL 2; A3: G*(i,j)`2 <= q`2 & q`2 <= G*(i,j+1)`2 by A1,A2,Th17; A4: p = |[q`1,q`2]| by EUCLID:53; G*(i,j)`1 <= q`1 & q`1 <= G*(i+1,j)`1 by A1,A2,Th17; hence p in A by A4,A3; end; hence cell(G,i,j) c= A by TARSKI:def 3; now let p be set; assume p in A; then consider r,s such that A5: |[r,s]| = p and A6: G*(i,j)`1 <= r & r <= G*(i+1,j)`1 & G*(i,j)`2 <= s & s <= G*(i,j+ 1)`2; reconsider q=p as Point of TOP-REAL 2 by A5; r = q`1 & s = q`2 by A5,EUCLID:52; hence p in cell(G,i,j) by A1,A6,Th17; end; hence thesis by TARSKI:def 3; end; theorem Th19: 1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G & p in Values G & p in cell(G,i,j) implies p = G*(i,j) or p = G*(i,j+1) or p = G*(i+1,j+1) or p = G*(i+1,j) proof assume that A1: 1 <= i and A2: i+1 <= len G and A3: 1 <= j and A4: j+1 <= width G and A5: p in Values G and A6: p in cell(G,i,j); A7: Values G = { G*(k,l) where k,l is Element of NAT: [k,l] in Indices G } by MATRIX_1:45; A8: i < len G by A2,NAT_1:13; A9: j < width G by A4,NAT_1:13; consider k,l being Element of NAT such that A10: p = G*(k,l) and A11: [k,l] in Indices G by A5,A7; A12: 1 <= k by A11,MATRIX_1:38; A13: l <= width G by A11,MATRIX_1:38; A14: 1 <= l by A11,MATRIX_1:38; A15: k <= len G by A11,MATRIX_1:38; A16: 1 <= j+1 by NAT_1:11; A17: now assume A18: l <> j & l <> j+1; per cases by A18,NAT_1:9; suppose l < j; then G*(k,l)`2 < G*(k,j)`2 by A9,A12,A15,A14,GOBOARD5:4; then G*(k,l)`2 < G*(1,j)`2 by A3,A9,A12,A15,GOBOARD5:1; then G*(k,l)`2 < G*(i,j)`2 by A1,A3,A8,A9,GOBOARD5:1; hence contradiction by A1,A2,A3,A4,A6,A10,Th17; end; suppose j+1 < l; then G*(k,j+1)`2 < G*(k,l)`2 by A16,A12,A15,A13,GOBOARD5:4; then G*(1,j+1)`2 < G*(k,l)`2 by A4,A16,A12,A15,GOBOARD5:1; then G*(i,j+1)`2 < G*(k,l)`2 by A1,A4,A8,A16,GOBOARD5:1; hence contradiction by A1,A2,A3,A4,A6,A10,Th17; end; end; A19: 1 <= i+1 by NAT_1:11; now assume A20: k <> i & k <> i+1; per cases by A20,NAT_1:9; suppose k < i; then G*(k,l)`1 < G*(i,l)`1 by A8,A12,A14,A13,GOBOARD5:3; then G*(k,l)`1 < G*(i,1)`1 by A1,A8,A14,A13,GOBOARD5:2; then G*(k,l)`1 < G*(i,j)`1 by A1,A3,A8,A9,GOBOARD5:2; hence contradiction by A1,A2,A3,A4,A6,A10,Th17; end; suppose i+1 < k; then G*(i+1,l)`1 < G*(k,l)`1 by A19,A15,A14,A13,GOBOARD5:3; then G*(i+1,1)`1 < G*(k,l)`1 by A2,A19,A14,A13,GOBOARD5:2; then G*(i+1,j)`1 < G*(k,l)`1 by A2,A3,A9,A19,GOBOARD5:2; hence contradiction by A1,A2,A3,A4,A6,A10,Th17; end; end; hence thesis by A10,A17; end; theorem Th20: 1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G implies G*(i,j) in cell(G,i,j) & G*(i,j+1) in cell(G,i,j) & G*(i+1,j+1) in cell(G,i,j) & G*(i+1 ,j) in cell(G,i,j) proof assume that A1: 1 <= i and A2: i+1 <= len G and A3: 1 <= j and A4: j+1 <= width G; A5: i < i+1 & j < width G by A4,NAT_1:13; then A6: G*(i,j)`1 <= G*(i+1,j)`1 by A1,A2,A3,GOBOARD5:3; A7: G*(i,j)`1 <= G*(i+1,j)`1 by A1,A2,A3,A5,GOBOARD5:3; A8: j < j+1 & i < len G by A2,NAT_1:13; then A9: G*(i,j)`2 <= G*(i,j+1)`2 by A1,A3,A4,GOBOARD5:4; A10: G*(i+1,j+1)`1 = G*(i+1,j)`1 by A1,A2,A3,A4,Th16; then A11: G*(i,j)`1 <= G*(i+1,j+1)`1 by A1,A2,A3,A5,GOBOARD5:3; G*(i,j)`2 <= G*(i,j+1)`2 by A1,A3,A4,A8,GOBOARD5:4; hence G*(i,j) in cell(G,i,j) by A1,A2,A3,A4,A6,Th17; A12: G*(i,j)`1 = G*(i,j+1)`1 by A1,A2,A3,A4,Th16; then G*(i,j+1)`1 <= G*(i+1,j)`1 by A1,A2,A3,A5,GOBOARD5:3; hence G*(i,j+1) in cell(G,i,j) by A1,A2,A3,A4,A12,A9,Th17; A13: G*(i+1,j+1)`2 = G*(i,j+1)`2 by A1,A2,A3,A4,Th16; then G*(i,j)`2 <= G*(i+1,j+1)`2 by A1,A3,A4,A8,GOBOARD5:4; hence G*(i+1,j+1) in cell(G,i,j) by A1,A2,A3,A4,A10,A11,A13,Th17; A14: G*(i,j)`2 = G*(i+1,j)`2 by A1,A2,A3,A4,Th16; then G*(i+1,j)`2 <= G*(i,j+1)`2 by A1,A3,A4,A8,GOBOARD5:4; hence thesis by A1,A2,A3,A4,A7,A14,Th17; end; theorem Th21: 1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G & p in Values G & p in cell(G,i,j) implies p is_extremal_in cell(G,i,j) proof assume that A1: 1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G and A2: p in Values G and A3: p in cell(G,i,j); for a,b being Point of TOP-REAL 2 st p in LSeg(a,b) & LSeg(a,b) c= cell( G,i,j) holds p = a or p = b proof let a,b be Point of TOP-REAL 2 such that A4: p in LSeg(a,b) and A5: LSeg(a,b) c= cell(G,i,j); A6: a in LSeg(a,b) by RLTOPSP1:68; A7: b in LSeg(a,b) by RLTOPSP1:68; assume that A8: a <> p and A9: b <> p; per cases by A1,A2,A3,Th19; suppose A10: p = G*(i,j); then A11: p`2 <= b`2 by A1,A5,A7,Th17; A12: p`1 <= a`1 by A1,A5,A6,A10,Th17; A13: p`1 <= b`1 by A1,A5,A7,A10,Th17; A14: p`2 <= a`2 by A1,A5,A6,A10,Th17; now per cases; suppose A15: a`1 <= b`1 & a`2 <= b`2; then a`2 <= p`2 by A4,TOPREAL1:4; then A16: a`2 = p`2 by A14,XXREAL_0:1; a`1 <= p`1 by A4,A15,TOPREAL1:3; then a`1 = p`1 by A12,XXREAL_0:1; hence contradiction by A8,A16,TOPREAL3:6; end; suppose A17: a`1 <= b`1 & b`2 < a`2; then b`2 <= p`2 by A4,TOPREAL1:4; then A18: b`2 = p`2 by A11,XXREAL_0:1; A19: a`1 <= p`1 by A4,A17,TOPREAL1:3; then A20: a`1 = p`1 by A12,XXREAL_0:1; then a`2 <> p`2 by A8,TOPREAL3:6; then LSeg(a,b) is vertical by A4,A6,A12,A19,SPPOL_1:18,XXREAL_0:1; then a`1 = b`1 by SPPOL_1:16; hence contradiction by A9,A20,A18,TOPREAL3:6; end; suppose A21: b`1 < a`1 & a`2 <= b`2; then a`2 <= p`2 by A4,TOPREAL1:4; then A22: a`2 = p`2 by A14,XXREAL_0:1; A23: b`1 <= p`1 by A4,A21,TOPREAL1:3; then A24: b`1 = p`1 by A13,XXREAL_0:1; then b`2 <> p`2 by A9,TOPREAL3:6; then LSeg(a,b) is vertical by A4,A7,A13,A23,SPPOL_1:18,XXREAL_0:1; then a`1 = b`1 by SPPOL_1:16; hence contradiction by A8,A24,A22,TOPREAL3:6; end; suppose A25: b`1 < a`1 & b`2 < a`2; then b`2 <= p`2 by A4,TOPREAL1:4; then A26: b`2 = p`2 by A11,XXREAL_0:1; b`1 <= p`1 by A4,A25,TOPREAL1:3; then b`1 = p`1 by A13,XXREAL_0:1; hence contradiction by A9,A26,TOPREAL3:6; end; end; hence contradiction; end; suppose A27: p = G*(i,j+1); then A28: b`2 <= p`2 by A1,A5,A7,Th17; A29: p`1 = G*(i,j)`1 by A1,A27,Th16; then A30: p`1 <= a`1 by A1,A5,A6,Th17; A31: p`1 <= b`1 by A1,A5,A7,A29,Th17; A32: a`2 <= p`2 by A1,A5,A6,A27,Th17; now per cases; suppose A33: a`1 <= b`1 & a`2 <= b`2; then p`2 <= b`2 by A4,TOPREAL1:4; then A34: b`2 = p`2 by A28,XXREAL_0:1; A35: a`1 <= p`1 by A4,A33,TOPREAL1:3; then A36: a`1 = p`1 by A30,XXREAL_0:1; then a`2 <> p`2 by A8,TOPREAL3:6; then LSeg(a,b) is vertical by A4,A6,A30,A35,SPPOL_1:18,XXREAL_0:1; then a`1 = b`1 by SPPOL_1:16; hence contradiction by A9,A36,A34,TOPREAL3:6; end; suppose A37: a`1 <= b`1 & b`2 < a`2; then p`2 <= a`2 by A4,TOPREAL1:4; then A38: a`2 = p`2 by A32,XXREAL_0:1; a`1 <= p`1 by A4,A37,TOPREAL1:3; then a`1 = p`1 by A30,XXREAL_0:1; hence contradiction by A8,A38,TOPREAL3:6; end; suppose A39: b`1 < a`1 & a`2 <= b`2; then p`2 <= b`2 by A4,TOPREAL1:4; then A40: b`2 = p`2 by A28,XXREAL_0:1; b`1 <= p`1 by A4,A39,TOPREAL1:3; then b`1 = p`1 by A31,XXREAL_0:1; hence contradiction by A9,A40,TOPREAL3:6; end; suppose A41: b`1 < a`1 & b`2 < a`2; then p`2 <= a`2 by A4,TOPREAL1:4; then A42: a`2 = p`2 by A32,XXREAL_0:1; A43: b`1 <= p`1 by A4,A41,TOPREAL1:3; then A44: b`1 = p`1 by A31,XXREAL_0:1; then b`2 <> p`2 by A9,TOPREAL3:6; then LSeg(a,b) is vertical by A4,A7,A31,A43,SPPOL_1:18,XXREAL_0:1; then a`1 = b`1 by SPPOL_1:16; hence contradiction by A8,A44,A42,TOPREAL3:6; end; end; hence contradiction; end; suppose A45: p = G*(i+1,j+1); then A46: p`1 = G*(i+1,j)`1 by A1,Th16; then A47: a`1 <= p`1 by A1,A5,A6,Th17; A48: p`2 = G*(i,j+1)`2 by A1,A45,Th16; then A49: b`2 <= p`2 by A1,A5,A7,Th17; A50: b`1 <= p`1 by A1,A5,A7,A46,Th17; A51: a`2 <= p`2 by A1,A5,A6,A48,Th17; now per cases; suppose A52: a`1 <= b`1 & a`2 <= b`2; then p`2 <= b`2 by A4,TOPREAL1:4; then A53: b`2 = p`2 by A49,XXREAL_0:1; p`1 <= b`1 by A4,A52,TOPREAL1:3; then b`1 = p`1 by A50,XXREAL_0:1; hence contradiction by A9,A53,TOPREAL3:6; end; suppose A54: a`1 <= b`1 & b`2 < a`2; then p`2 <= a`2 by A4,TOPREAL1:4; then A55: a`2 = p`2 by A51,XXREAL_0:1; A56: p`1 <= b`1 by A4,A54,TOPREAL1:3; then A57: b`1 = p`1 by A50,XXREAL_0:1; then b`2 <> p`2 by A9,TOPREAL3:6; then LSeg(a,b) is vertical by A4,A7,A50,A56,SPPOL_1:18,XXREAL_0:1; then a`1 = b`1 by SPPOL_1:16; hence contradiction by A8,A57,A55,TOPREAL3:6; end; suppose A58: b`1 < a`1 & a`2 <= b`2; then p`2 <= b`2 by A4,TOPREAL1:4; then A59: b`2 = p`2 by A49,XXREAL_0:1; A60: p`1 <= a`1 by A4,A58,TOPREAL1:3; then A61: a`1 = p`1 by A47,XXREAL_0:1; then a`2 <> p`2 by A8,TOPREAL3:6; then LSeg(a,b) is vertical by A4,A6,A47,A60,SPPOL_1:18,XXREAL_0:1; then a`1 = b`1 by SPPOL_1:16; hence contradiction by A9,A61,A59,TOPREAL3:6; end; suppose A62: b`1 < a`1 & b`2 < a`2; then p`2 <= a`2 by A4,TOPREAL1:4; then A63: a`2 = p`2 by A51,XXREAL_0:1; p`1 <= a`1 by A4,A62,TOPREAL1:3; then a`1 = p`1 by A47,XXREAL_0:1; hence contradiction by A8,A63,TOPREAL3:6; end; end; hence contradiction; end; suppose A64: p = G*(i+1,j); then A65: p`2 = G*(i,j)`2 by A1,Th16; then A66: p`2 <= b`2 by A1,A5,A7,Th17; A67: a`1 <= p`1 by A1,A5,A6,A64,Th17; A68: b`1 <= p`1 by A1,A5,A7,A64,Th17; A69: p`2 <= a`2 by A1,A5,A6,A65,Th17; now per cases; suppose A70: a`1 <= b`1 & a`2 <= b`2; then a`2 <= p`2 by A4,TOPREAL1:4; then A71: a`2 = p`2 by A69,XXREAL_0:1; A72: p`1 <= b`1 by A4,A70,TOPREAL1:3; then A73: b`1 = p`1 by A68,XXREAL_0:1; then b`2 <> p`2 by A9,TOPREAL3:6; then LSeg(a,b) is vertical by A4,A7,A68,A72,SPPOL_1:18,XXREAL_0:1; then a`1 = b`1 by SPPOL_1:16; hence contradiction by A8,A73,A71,TOPREAL3:6; end; suppose A74: a`1 <= b`1 & b`2 < a`2; then b`2 <= p`2 by A4,TOPREAL1:4; then A75: b`2 = p`2 by A66,XXREAL_0:1; p`1 <= b`1 by A4,A74,TOPREAL1:3; then b`1 = p`1 by A68,XXREAL_0:1; hence contradiction by A9,A75,TOPREAL3:6; end; suppose A76: b`1 < a`1 & a`2 <= b`2; then a`2 <= p`2 by A4,TOPREAL1:4; then A77: a`2 = p`2 by A69,XXREAL_0:1; p`1 <= a`1 by A4,A76,TOPREAL1:3; then a`1 = p`1 by A67,XXREAL_0:1; hence contradiction by A8,A77,TOPREAL3:6; end; suppose A78: b`1 < a`1 & b`2 < a`2; then b`2 <= p`2 by A4,TOPREAL1:4; then A79: b`2 = p`2 by A66,XXREAL_0:1; A80: p`1 <= a`1 by A4,A78,TOPREAL1:3; then A81: a`1 = p`1 by A67,XXREAL_0:1; then a`2 <> p`2 by A8,TOPREAL3:6; then LSeg(a,b) is vertical by A4,A6,A67,A80,SPPOL_1:18,XXREAL_0:1; then a`1 = b`1 by SPPOL_1:16; hence contradiction by A9,A81,A79,TOPREAL3:6; end; end; hence contradiction; end; end; hence thesis by A3,SPPOL_1:def 1; end; theorem Th22: 2 <= len G & 2 <= width G & f is_sequence_on G & 1 <= k & k+1 <= len f implies ex i,j st 1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G & LSeg( f,k) c= cell(G,i,j) proof assume that A1: 2 <= len G and A2: 2 <= width G and A3: f is_sequence_on G and A4: 1 <= k & k+1 <= len f; consider i1,j1,i2,j2 being Element of NAT such that A5: [i1,j1] in Indices G and A6: f/.k = G*(i1,j1) and A7: [i2,j2] in Indices G and A8: f/.(k+1) = G*(i2,j2) and A9: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A3,A4,JORDAN8:3; A10: LSeg(f,k) = LSeg(f/.k, f/.(k+1)) by A4,TOPREAL1:def 3; A11: 1 <= i2 by A7,MATRIX_1:38; A12: 1 <= i1 by A5,MATRIX_1:38; A13: 1 <= j2 by A7,MATRIX_1:38; A14: 1 <= j1 by A5,MATRIX_1:38; A15: i2 <= len G by A7,MATRIX_1:38; A16: i1 <= len G by A5,MATRIX_1:38; A17: j2 <= width G by A7,MATRIX_1:38; A18: j1 <= width G by A5,MATRIX_1:38; per cases by A9; suppose A19: i1 = i2 & j1+1 = j2; then A20: j1 < width G by A17,XREAL_1:145; now per cases by A16,XXREAL_0:1; suppose A21: i1 < len G; take i1,j1; A22: i1+1 <= len G by A21,NAT_1:13; LSeg(f,k) c= cell(G,i1,j1) by A10,A6,A8,A12,A16,A14,A17,A19,GOBOARD5:19 ,XREAL_1:145; hence thesis by A12,A14,A17,A19,A22; end; suppose A23: i1 = len G; take i19=i1-'1,j1; 2-1 <= 2-'1 & 2-'1 <= i19 by A1,A23,NAT_D:42,XREAL_0:def 2; then A24: 1 <= i19 by XXREAL_0:2; A25: i19+1 = i1 by A12,XREAL_1:235; then i19 < len G by A16,NAT_1:13; then LSeg(f,k) c= cell(G,i19,j1) by A10,A6,A8,A14,A19,A20,A25, GOBOARD5:18; hence thesis by A16,A14,A17,A19,A24,A25; end; end; hence thesis; end; suppose A26: i1+1 = i2 & j1 = j2; then A27: i1 < len G by A15,XREAL_1:145; now per cases by A18,XXREAL_0:1; suppose A28: j1 < width G; take i1,j1; A29: j1+1 <= width G by A28,NAT_1:13; LSeg(f,k) c= cell(G,i1,j1) by A10,A6,A8,A12,A14,A18,A15,A26,GOBOARD5:22 ,XREAL_1:145; hence thesis by A12,A14,A15,A26,A29; end; suppose A30: j1 = width G; take i1,j19=j1-'1; 2-1 <= 2-'1 & 2-'1 <= j19 by A2,A30,NAT_D:42,XREAL_0:def 2; then A31: 1 <= j19 by XXREAL_0:2; A32: j19+1=j1 by A14,XREAL_1:235; then j19 < width G by A30,NAT_1:13; then LSeg(f,k) c= cell(G,i1,j19) by A10,A6,A8,A12,A26,A27,A32, GOBOARD5:21; hence thesis by A12,A18,A15,A26,A31,A32; end; end; hence thesis; end; suppose A33: i1 = i2+1 & j1 = j2; then A34: i2 < len G by A16,XREAL_1:145; now per cases by A18,XXREAL_0:1; suppose A35: j1 < width G; take i2,j1; A36: j1+1 <= width G by A35,NAT_1:13; LSeg(f,k) c= cell(G,i2,j1) by A10,A6,A8,A16,A11,A13,A17,A33,GOBOARD5:22 ,XREAL_1:145; hence thesis by A16,A14,A11,A33,A36; end; suppose A37: j1 = width G; take i2,j19=j1-'1; 2-1 <= 2-'1 & 2-'1 <= j19 by A2,A37,NAT_D:42,XREAL_0:def 2; then A38: 1 <= j19 by XXREAL_0:2; A39: j19+1=j1 by A14,XREAL_1:235; then j19 < width G by A37,NAT_1:13; then LSeg(f,k) c= cell(G,i2,j19) by A10,A6,A8,A11,A33,A34,A39, GOBOARD5:21; hence thesis by A16,A18,A11,A33,A38,A39; end; end; hence thesis; end; suppose A40: i1 = i2 & j1 = j2+1; then A41: j2 < width G by A18,XREAL_1:145; now per cases by A16,XXREAL_0:1; suppose A42: i1 < len G; take i1,j2; A43: i1+1 <= len G by A42,NAT_1:13; LSeg(f,k) c= cell(G,i1,j2) by A10,A6,A8,A18,A11,A15,A13,A40,GOBOARD5:19 ,XREAL_1:145; hence thesis by A12,A18,A13,A40,A43; end; suppose A44: i1 = len G; take i19=i1-'1,j2; 2-1 <= 2-'1 & 2-'1 <= i19 by A1,A44,NAT_D:42,XREAL_0:def 2; then A45: 1 <= i19 by XXREAL_0:2; A46: i19+1 = i1 by A12,XREAL_1:235; then i19 < len G by A16,NAT_1:13; then LSeg(f,k) c= cell(G,i19,j2) by A10,A6,A8,A13,A40,A41,A46, GOBOARD5:18; hence thesis by A16,A18,A13,A40,A45,A46; end; end; hence thesis; end; end; theorem Th23: 2 <= len G & 2 <= width G & f is_sequence_on G & 1 <= k & k+1 <= len f & p in Values G & p in LSeg(f,k) implies p = f/.k or p = f/.(k+1) proof assume that A1: 2 <= len G & 2 <= width G & f is_sequence_on G and A2: 1 <= k & k+1 <= len f and A3: p in Values G and A4: p in LSeg(f,k); A5: LSeg(f,k) = LSeg(f/.k, f/.(k+1)) by A2,TOPREAL1:def 3; consider i,j such that A6: 1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G and A7: LSeg(f,k) c= cell(G,i,j) by A1,A2,Th22; p is_extremal_in cell(G,i,j) by A3,A4,A6,A7,Th21; hence thesis by A4,A7,A5,SPPOL_1:def 1; end; theorem [i,j] in Indices G & 1 <= k & k <= width G implies G*(i,j)`1 <= G* ( len G,k)`1 proof assume that A1: [i,j] in Indices G and A2: 1 <= k & k <= width G; A3: 1 <= i by A1,MATRIX_1:38; A4: i <= len G by A1,MATRIX_1:38; then A5: i < len G or i = len G by XXREAL_0:1; 1 <= j & j <= width G by A1,MATRIX_1:38; then G*(i,j)`1 = G*(i,1)`1 by A3,A4,GOBOARD5:2 .= G*(i,k)`1 by A2,A3,A4,GOBOARD5:2; hence thesis by A2,A3,A5,GOBOARD5:3; end; theorem [i,j] in Indices G & 1 <= k & k <= len G implies G*(i,j)`2 <= G* (k, width G)`2 proof assume that A1: [i,j] in Indices G and A2: 1 <= k & k <= len G; A3: 1 <= j by A1,MATRIX_1:38; A4: j <= width G by A1,MATRIX_1:38; then A5: j < width G or j = width G by XXREAL_0:1; 1 <= i & i <= len G by A1,MATRIX_1:38; then G*(i,j)`2 = G*(1,j)`2 by A3,A4,GOBOARD5:1 .= G*(k,j)`2 by A2,A3,A4,GOBOARD5:1; hence thesis by A2,A3,A5,GOBOARD5:4; end; theorem Th26: f is_sequence_on G & f is special & L~g c= L~f & 1 <= k & k+1 <= len f implies for A being Subset of TOP-REAL 2 st A = right_cell(f,k,G)\L~g or A = left_cell(f,k,G)\L~g holds A is connected proof assume that A1: f is_sequence_on G and A2: f is special and A3: L~g c= L~f and A4: 1 <= k & k+1 <= len f; let A be Subset of TOP-REAL 2 such that A5: A = right_cell(f,k,G)\L~g or A = left_cell(f,k,G)\L~g; per cases by A5; suppose A6: A = right_cell(f,k,G)\L~g; Int right_cell(f,k,G) misses L~f by A1,A2,A4,Th15; then Int right_cell(f,k,G) misses L~g by A3,XBOOLE_1:63; then A7: Int right_cell(f,k,G) c= (L~g)` by SUBSET_1:23; A c= right_cell(f,k,G) by A6,XBOOLE_1:36; then A8: A c= Cl Int right_cell(f,k,G) by A1,A4,Th11; A9: A = right_cell(f,k,G) /\ (L~g)` by A6,SUBSET_1:13; Int right_cell(f,k,G) is convex & Int right_cell(f,k,G) c= right_cell(f,k,G) by A1,A4,Th10,TOPS_1:16; hence thesis by A9,A7,A8,CONNSP_1:18,XBOOLE_1:19; end; suppose A10: A = left_cell(f,k,G)\L~g; Int left_cell(f,k,G) misses L~f by A1,A2,A4,Th15; then Int left_cell(f,k,G) misses L~g by A3,XBOOLE_1:63; then A11: Int left_cell(f,k,G) c= (L~g)` by SUBSET_1:23; A c= left_cell(f,k,G) by A10,XBOOLE_1:36; then A12: A c= Cl Int left_cell(f,k,G) by A1,A4,Th11; A13: A = left_cell(f,k,G) /\ (L~g)` by A10,SUBSET_1:13; Int left_cell(f,k,G) is convex & Int left_cell(f,k,G) c= left_cell (f,k,G) by A1,A4,Th10,TOPS_1:16; hence thesis by A13,A11,A12,CONNSP_1:18,XBOOLE_1:19; end; end; theorem Th27: for f being non constant standard special_circular_sequence st f is_sequence_on G for k st 1 <= k & k+1 <= len f holds right_cell(f,k,G)\L~f c= RightComp f & left_cell(f,k,G)\L~f c= LeftComp f proof let f be non constant standard special_circular_sequence such that A1: f is_sequence_on G; let k such that A2: 1 <= k & k+1 <= len f; A3: Int right_cell(f,k,G) <> {} by A1,A2,Th9; set rc = right_cell(f,k,G)\L~f; rc \/ L~f = right_cell(f,k,G) \/ L~f by XBOOLE_1:39; then Int right_cell(f,k,G) c= right_cell(f,k,G) & right_cell(f,k,G) c= rc \/ L~f by TOPS_1:16,XBOOLE_1:7; then A4: Int right_cell(f,k,G) c= rc \/ L~f by XBOOLE_1:1; set lc = left_cell(f,k,G)\L~f; rc = right_cell(f,k,G) /\ (L~f)` by SUBSET_1:13; then A5: RightComp f is_a_component_of (L~f)` & rc c= (L~f)` by GOBOARD9:def 2 ,XBOOLE_1:17; rc c= right_cell(f,k,G) & right_cell(f,k,G) c= right_cell(f,k) by A1,A2, GOBRD13:33,XBOOLE_1:36; then rc c= right_cell(f,k) by XBOOLE_1:1; then A6: Int rc c= Int right_cell(f,k) by TOPS_1:19; Int right_cell(f,k) c= RightComp f by A2,GOBOARD9:25; then A7: Int rc c= RightComp f by A6,XBOOLE_1:1; Int right_cell(f,k,G) misses L~f by A1,A2,Th15; then A8: Int Int right_cell(f,k,G) c= Int rc by A4,TOPS_1:19,XBOOLE_1:73; Int right_cell(f,k,G) c= rc by A1,A2,A4,Th15,XBOOLE_1:73; then A9: rc meets Int rc by A3,A8,XBOOLE_1:68; rc is connected by A1,A2,Th26; hence right_cell(f,k,G)\L~f c= RightComp f by A7,A9,A5,GOBOARD9:4; lc = left_cell(f,k,G) /\ (L~f)` by SUBSET_1:13; then A10: LeftComp f is_a_component_of (L~f)` & lc c= (L~f)` by GOBOARD9:def 1 ,XBOOLE_1:17; lc \/ L~f = left_cell(f,k,G) \/ L~f by XBOOLE_1:39; then Int left_cell(f,k,G) c= left_cell(f,k,G) & left_cell(f,k,G) c= lc \/ L~ f by TOPS_1:16,XBOOLE_1:7; then A11: Int left_cell(f,k,G) c= lc \/ L~f by XBOOLE_1:1; lc c= left_cell(f,k,G) & left_cell(f,k,G) c= left_cell(f,k) by A1,A2, GOBRD13:33,XBOOLE_1:36; then lc c= left_cell(f,k) by XBOOLE_1:1; then A12: Int lc c= Int left_cell(f,k) by TOPS_1:19; Int left_cell(f,k) c= LeftComp f by A2,GOBOARD9:21; then A13: Int lc c= LeftComp f by A12,XBOOLE_1:1; A14: Int left_cell(f,k,G) <> {} by A1,A2,Th9; Int left_cell(f,k,G) misses L~f by A1,A2,Th15; then A15: Int Int left_cell(f,k,G) c= Int lc by A11,TOPS_1:19,XBOOLE_1:73; Int left_cell(f,k,G) c= lc by A1,A2,A11,Th15,XBOOLE_1:73; then A16: lc meets Int lc by A14,A15,XBOOLE_1:68; lc is connected by A1,A2,Th26; hence thesis by A13,A16,A10,GOBOARD9:4; end; begin :: Cages reserve C for compact non vertical non horizontal non empty Subset of TOP-REAL 2, l, m, i1, i2, j1, j2 for Element of NAT; theorem Th28: for n being Nat ex i st 1 <= i & i+1 <= len Gauge(C,n) & N-min C in cell(Gauge(C,n),i,width Gauge(C,n)-'1) & N-min C <> Gauge(C,n)*(i,width Gauge(C,n)-'1) proof let n be Nat; set G = Gauge(C,n); defpred P[Nat] means 1 <= $1 & $1 < len G & G*($1,(width G)-'1)`1 < (N-min C )`1; A1: for k be Nat st P[k] holds k <= len G; A2: len G = width G by JORDAN8:def 1; (NW-corner C)`1 <= (N-min C)`1 by PSCOMP_1:38; then A3: W-bound C <= (N-min C)`1 by EUCLID:52; A4: len G >= 4 by JORDAN8:10; then A5: (len G)-'1 <= len G & 2 <= len G by NAT_D:35,XXREAL_0:2; A6: 1 < len G by A4,XXREAL_0:2; then A7: 1 <= (len G)-'1 by NAT_D:49; A8: n in NAT & len G = width G by JORDAN8:def 1,ORDINAL1:def 12; then G*(2,(width G)-'1)`1 = W-bound C by A7,JORDAN8:11,NAT_D:35; then G*(1,(width G)-'1)`1 < W-bound C by A2,A7,A5,GOBOARD5:3; then G*(1,(width G)-'1)`1 < (N-min C)`1 by A3,XXREAL_0:2; then A9: ex k be Nat st P[k] by A6; ex i being Nat st P[i] & for n be Nat st P[n] holds n <= i from NAT_1: sch 6(A1,A9); then consider i being Nat such that A10: 1 <= i and A11: i < len G and A12: G*(i,(width G)-'1)`1 < (N-min C)`1 and A13: for n be Nat st P[n] holds n <= i; reconsider i as Element of NAT by ORDINAL1:def 12; A14: 1 <= i+1 & i < i+1 by NAT_1:12,13; A15: (N-min C)`2 = N-bound C by EUCLID:52; A16: i+1 <= len G by A11,NAT_1:13; then A17: (N-min C)`2 = G* (i+1,(width G)-'1)`2 by A8,A15,JORDAN8:14,NAT_1:12; now assume i+1 = len G; then len G-'1 = i by NAT_D:34; then A18: G*(i,(width G)-'1)`1 = E-bound C by A8,A7,JORDAN8:12,NAT_D:35; (NE-corner C)`1 >= (N-min C)`1 by PSCOMP_1:38; hence contradiction by A12,A18,EUCLID:52; end; then i+1 < len G by A16,XXREAL_0:1; then A19: (N-min C)`1 <= G*(i+1,(width G)-'1)`1 by A13,A14; G*(i,(width G)-'1)`2 = (N-min C)`2 by A8,A10,A11,A15,JORDAN8:14; then A20: N-min C in LSeg(G*(i,(width G)-'1),G*(i+1,(width G)-'1)) by A12,A17,A19, GOBOARD7:8; take i; thus 1 <= i & i+1 <= len G by A10,A11,NAT_1:13; LSeg(G*(i,(width G)-'1),G*(i+1,(width G)-'1)) c= cell(G,i,(width G)-'1) by A2,A7,A10,A11,GOBOARD5:22,NAT_D:35; hence N-min C in cell(G,i,(width G)-'1) by A20; thus thesis by A12; end; theorem Th29: for n, i1, i2 being Nat holds 1 <= i1 & i1+1 <= len Gauge(C,n) & N-min C in cell(Gauge(C,n),i1,width Gauge(C,n)-'1) & N-min C <> Gauge(C,n)*(i1, width Gauge(C,n)-'1) & 1 <= i2 & i2+1 <= len Gauge(C,n) & N-min C in cell(Gauge (C,n),i2,width Gauge(C,n)-'1) & N-min C <> Gauge(C,n)*(i2,width Gauge(C,n)-'1) implies i1 = i2 proof let n, i1, i2 be Nat; A1: i1 in NAT by ORDINAL1:def 12; set G = Gauge(C,n), j = width G-'1; A2: i2 in NAT by ORDINAL1:def 12; A3: 2|^n >= n+1 by NEWTON:85; A4: 1+ (n+3) > 1+0 by XREAL_1:6; A5: len G = width G by JORDAN8:def 1; A6: len G = 2|^n+3 by JORDAN8:def 1; then A7: len G >= n+1+3 by A3,XREAL_1:6; then len G > 1 by A4,XXREAL_0:2; then A8: len G >= 1+1 by NAT_1:13; then A9: 1 <= j by A5,JORDAN5B:2; A10: j+1 = len G by A5,A7,A4,XREAL_1:235,XXREAL_0:2; then A11: j < len G by NAT_1:13; assume that A12: 1 <= i1 and A13: i1+1 <= len G and A14: N-min C in cell(G,i1,j) and A15: N-min C <> G*(i1,j) and A16: 1 <= i2 and A17: i2+1 <= len G and A18: N-min C in cell(G,i2,j) and A19: N-min C <> G*(i2,j) and A20: i1 <> i2; A21: cell(G,i1,j) meets cell(G,i2,j) by A14,A18,XBOOLE_0:3; A22: i1 < len G by A13,NAT_1:13; A23: i2 < len G by A17,NAT_1:13; per cases by A20,XXREAL_0:1; suppose A24: i1 < i2; then A25: i2-'i1+i1 = i2 by XREAL_1:235; then i2-'i1 <= 1 by A1,A23,A5,A21,A9,A11,JORDAN8:7; then i2-'i1 < 1 or i2-'i1 = 1 by XXREAL_0:1; then i2-'i1 = 0 or i2-'i1 = 1 by NAT_1:14; then cell(G,i1,j) /\ cell(G,i2,j) = LSeg(G*(i2,j),G*(i2,j+1)) by A1,A22,A5 ,A8,A11,A24,A25,GOBOARD5:25,JORDAN5B:2; then A26: N-min C in LSeg(G*(i2,j),G*(i2,j+1)) by A14,A18,XBOOLE_0:def 4; 1 <= j+1 by NAT_1:12; then A27: [i2,j+1] in Indices G by A16,A23,A5,A10,MATRIX_1:36; set y2 = (S-bound C)+(((N-bound C)-(S-bound C))/(2|^n))*(j-1); set y1 = (S-bound C)+(((N-bound C)-(S-bound C))/(2|^n))*(j-2); set x = (W-bound C)+(((E-bound C)-(W-bound C))/(2|^n))*(i2-2); j = (2|^n+2+1)-'1 by A6,JORDAN8:def 1 .= (2|^n+2) by NAT_D:34; then A28: (((N-bound C)-(S-bound C))/(2|^n))*(j-2) = (N-bound C)-(S-bound C) by A3, XCMPLX_1:87; [i2,j] in Indices G by A16,A23,A5,A9,A11,MATRIX_1:36; then A29: G*(i2,j) = |[x,y1]| by JORDAN8:def 1; then A30: G*(i2,j)`1 = x by EUCLID:52; j+1-(1+1) = j-1; then G*(i2,j+1) = |[x,y2]| by A27,JORDAN8:def 1; then G*(i2,j+1)`1 = x by EUCLID:52; then LSeg(G*(i2,j),G*(i2,j+1)) is vertical by A30,SPPOL_1:16; then (N-min C)`1 = G*(i2,j)`1 by A26,SPPOL_1:41; hence contradiction by A19,A29,A30,A28,EUCLID:52; end; suppose A31: i2 < i1; then A32: i1-'i2+i2 = i1 by XREAL_1:235; then i1-'i2 <= 1 by A2,A22,A5,A21,A9,A11,JORDAN8:7; then i1-'i2 < 1 or i1-'i2 = 1 by XXREAL_0:1; then i1-'i2 = 0 or i1-'i2 = 1 by NAT_1:14; then cell(G,i2,j) /\ cell(G,i1,j) = LSeg(G*(i1,j),G*(i1,j+1)) by A2,A23,A5 ,A8,A11,A31,A32,GOBOARD5:25,JORDAN5B:2; then A33: N-min C in LSeg(G*(i1,j),G*(i1,j+1)) by A14,A18,XBOOLE_0:def 4; 1 <= j+1 by NAT_1:12; then A34: [i1,j+1] in Indices G by A12,A22,A5,A10,MATRIX_1:36; set y2 = (S-bound C)+(((N-bound C)-(S-bound C))/(2|^n))*(j-1); set y1 = (S-bound C)+(((N-bound C)-(S-bound C))/(2|^n))*(j-2); set x = (W-bound C)+(((E-bound C)-(W-bound C))/(2|^n))*(i1-2); j = (2|^n+2+1)-'1 by A6,JORDAN8:def 1 .= (2|^n+2) by NAT_D:34; then A35: (((N-bound C)-(S-bound C))/(2|^n))*(j-2) = (N-bound C)-(S-bound C) by A3, XCMPLX_1:87; [i1,j] in Indices G by A12,A22,A5,A9,A11,MATRIX_1:36; then A36: G*(i1,j) = |[x,y1]| by JORDAN8:def 1; then A37: G*(i1,j)`1 = x by EUCLID:52; j+1-(1+1) = j-1; then G*(i1,j+1) = |[x,y2]| by A34,JORDAN8:def 1; then G*(i1,j+1)`1 = x by EUCLID:52; then LSeg(G*(i1,j),G*(i1,j+1)) is vertical by A37,SPPOL_1:16; then (N-min C)`1 = G*(i1,j)`1 by A33,SPPOL_1:41; hence contradiction by A15,A36,A37,A35,EUCLID:52; end; end; theorem Th30: for n being Nat for f being standard non constant special_circular_sequence st f is_sequence_on Gauge(C,n) & (for k st 1 <= k & k +1 <= len f holds left_cell(f,k,Gauge(C,n)) misses C & right_cell(f,k,Gauge(C,n )) meets C) & (ex i st 1 <= i & i+1 <= len Gauge(C,n) & f/.1 = Gauge(C,n)*(i, width Gauge(C,n)) & f/.2 = Gauge(C,n)*(i+1,width Gauge(C,n)) & N-min C in cell( Gauge(C,n),i,width Gauge(C,n)-'1) & N-min C <> Gauge(C,n)*(i,width Gauge(C,n)-' 1)) holds N-min L~f = f/.1 proof let n be Nat; set G = Gauge(C,n); let f be standard non constant special_circular_sequence such that A1: f is_sequence_on G and A2: for k st 1 <= k & k+1 <= len f holds left_cell(f,k,G) misses C & right_cell(f,k,G) meets C; N-min L~f in rng f by SPRECT_2:39; then consider m being Nat such that A3: m in dom f and A4: f.m = N-min L~f by FINSEQ_2:10; reconsider m as Element of NAT by ORDINAL1:def 12; consider i,j such that A5: [i,j] in Indices G and A6: f/.m = G*(i,j) by A1,A3,GOBOARD1:def 9; A7: f/.m = f.m by A3,PARTFUN1:def 6; A8: (N-min L~f)`2 = N-bound L~f by EUCLID:52; set W = W-bound C, S = S-bound C, E = E-bound C, N = N-bound C; given i9 being Element of NAT such that A9: 1 <= i9 and A10: i9+1 <= len G and A11: f/.1 = G*(i9,width G) and A12: f/.2 = G*(i9+1,width G) and A13: N-min C in cell(G,i9,width G-'1) and A14: N-min C <> G*(i9,width G-'1); A15: G*(i9,len G-'1) = |[G*(i9,len G-'1)`1,G*(i9,len G-'1)`2]| & (N-min C)`2 = N by EUCLID:52,53; G*(i,j) = |[W+((E-W)/(2|^n))*(i-2), S+((N-S)/(2|^n))*(j-2)]| by A5, JORDAN8:def 1; then A16: S+((N-S)/(2|^n))*(j-2) = N-bound L~f by A4,A7,A8,A6,EUCLID:52; N > S by JORDAN8:9; then 2|^n > 0 & N-S > 0 by NEWTON:83,XREAL_1:50; then A17: (N-S)/(2|^n) > 0 by XREAL_1:139; A18: (NW-corner L~f)`1 = W-bound L~f & (NE-corner L~f)`1 = E-bound L~f by EUCLID:52; A19: 1 <= i by A5,MATRIX_1:38; A20: (NW-corner L~f)`2 = N-bound L~f & (NE-corner L~f)`2 = N-bound L~f by EUCLID:52; A21: m <= len f by A3,FINSEQ_3:25; A22: 1 <= j by A5,MATRIX_1:38; len G = 2|^n+3 by JORDAN8:def 1; then A23: len G >= 3 by NAT_1:12; then A24: 1 < len G by XXREAL_0:2; then A25: 1 <= len G-'1 by NAT_D:49; then A26: len G-'1 < len G by NAT_D:51; A27: i <= len G by A5,MATRIX_1:38; A28: j <= width G by A5,MATRIX_1:38; then A29: G*(i,j)`2 = G* (1,j)`2 by A19,A27,A22,GOBOARD5:1; A30: len f > 4 by GOBOARD7:34; 1 in dom f by FINSEQ_5:6; then A31: f/.1 in L~f by A30,GOBOARD1:1,XXREAL_0:2; then A32: N-bound L~f >= (f/.1)`2 by PSCOMP_1:24; A33: len G = width G by JORDAN8:def 1; A34: i9 < len G by A10,NAT_1:13; then G*(i9,j)`2 = G*(1,j)`2 by A9,A22,A28,GOBOARD5:1; then G*(i,j)`2 <= G*(i9,len G)`2 by A9,A34,A33,A22,A28,A29,SPRECT_3:12; then A35: N-bound L~f = (f/.1)`2 by A11,A33,A4,A7,A8,A6,A32,XXREAL_0:1; [i9,len G] in Indices G by A9,A34,A33,A24,MATRIX_1:36; then G*(i9,len G)=|[W+((E-W)/(2|^n))*(i9-2),S+((N-S)/(2|^n))*(len G-2)]| by JORDAN8:def 1; then S+((N-S)/(2|^n))*(len G-2) = N-bound L~f by A11,A33,A35,EUCLID:52; then A36: len G-2 = j-2 by A17,A16,XCMPLX_1:5; then A37: G*(i9,len G)`1 = G*(i9,1)`1 by A9,A34,A33,A22,GOBOARD5:2; W-bound L~f <= (f/.1)`1 & (f/.1)`1 <= E-bound L~f by A31,PSCOMP_1:24; then f/.1 in LSeg(NW-corner L~f, NE-corner L~f) by A35,A18,A20,GOBOARD7:8; then f/.1 in LSeg(NW-corner L~f, NE-corner L~f) /\ L~f by A31,XBOOLE_0:def 4; then A38: (N-min L~f)`1 <= (f/.1)`1 by PSCOMP_1:39; then A39: i <= i9 by A9,A11,A33,A4,A7,A6,A27,A22,A36,GOBOARD5:3; then A40: i < len G by A34,XXREAL_0:2; then A41: i+1 <= len G by NAT_1:13; A42: len G-'1+1 = len G by A23,XREAL_1:235,XXREAL_0:2; then N-min C in { |[r9,s9]| where r9,s9 is Real: G*(i9,1)`1 <= r9 & r9 <= G* (i9+1,1)`1 & G*(1,len G-'1)`2 <= s9 & s9 <= G*(1,len G)`2 } by A9,A13,A34,A33 ,A25,A26,GOBRD11:32; then ex r9,s9 being Real st N-min C = |[r9,s9]| & G*(i9,1)`1 <= r9 & r9 <= G* (i9+1,1)`1 & G*(1,len G-'1)`2 <= s9 & s9 <= G*(1,len G)`2; then A43: (f/.1)`1 <= (N-min C)`1 by A11,A33,A37,EUCLID:52; then A44: (N-min L~f)`1 <= (N-min C)`1 by A38,XXREAL_0:2; A45: 1 <= m by A3,FINSEQ_3:25; A46: n in NAT by ORDINAL1:def 12; then A47: G*(i9,len G-'1)`2 = N by A9,A34,JORDAN8:14; A48: N-min C = |[(N-min C)`1,(N-min C)`2]| by EUCLID:53; A49: (NW-corner C)`2 = N & (NE-corner C)`2 = N by EUCLID:52; A50: (NW-corner C)`1 = W & (NE-corner C)`1 = E by EUCLID:52; A51: len G = width G by JORDAN8:def 1; G*(i9,len G-'1)`1 = G*(i9,1)`1 by A9,A34,A33,A25,A26,GOBOARD5:2; then A52: G*(i9,len G-'1)`1 < (N-min C)`1 by A11,A14,A33,A37,A43,A48,A15,A47, XXREAL_0:1; A53: G*(i,len G)`1 = G*(i,1)`1 by A19,A27,A22,A28,A36,GOBOARD5:2; per cases by A21,XXREAL_0:1; suppose m = len f; hence thesis by A4,A7,FINSEQ_6:def 1; end; suppose m < len f; then A54: m+1 <= len f by NAT_1:13; then consider i1,j1,i2,j2 being Element of NAT such that A55: [i1,j1] in Indices G & f/.m = G*(i1,j1) and A56: [i2,j2] in Indices G and A57: f/.(m+1) = G*(i2,j2) and A58: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A1,A45,JORDAN8:3; A59: right_cell(f,m,G) meets C by A2,A45,A54; then consider p being set such that A60: p in right_cell(f,m,G) and A61: p in C by XBOOLE_0:3; reconsider p as Point of TOP-REAL 2 by A60; A62: W <= p`1 & p`1 <= E by A61,PSCOMP_1:24; A63: (N-min C)`2 = N by EUCLID:52; then A64: p`2 <= (N-min C)`2 by A61,PSCOMP_1:24; A65: G*(1,len G-'1)`2 < G*(1,len G)`2 by A51,A24,A25,A26,GOBOARD5:4; A66: G*(1,len G-'1)`2 = N by A24,A46,JORDAN8:14; A67: j2 <= len G by A51,A56,MATRIX_1:38; now per cases by A5,A6,A36,A55,A58,GOBOARD1:5; suppose i = i2 & len G+1 = j2; hence thesis by A67,NAT_1:13; end; suppose A68: i+1 = i2 & len G = j2; A69: cell(G,i,len G-'1) = {|[r,s]| : G*(i,1)`1 <= r & r <= G*(i+1,1)`1 & G*(1,len G-'1)`2 <= s & s <= G*(1,len G-'1+1)`2 } by A51,A19,A25,A26,A40, GOBRD11:32; right_cell(f,m,G) = cell(G,i,len G-'1) by A1,A45,A5,A6,A36,A54,A56,A57 ,A68,GOBRD13:24; then consider r,s such that A70: p = |[r,s]| and G*(i,1)`1 <= r and A71: r <= G*(i+1,1)`1 and A72: G*(1,len G-'1)`2 <= s and s <= G*(1,len G-'1+1)`2 by A60,A69; p`2 = s by A70,EUCLID:52; then p`2 = N by A63,A64,A66,A72,XXREAL_0:1; then p in LSeg(NW-corner C, NE-corner C) by A50,A49,A62,GOBOARD7:8; then p in LSeg(NW-corner C, NE-corner C) /\ C by A61,XBOOLE_0:def 4; then A73: (N-min C)`1 <= p`1 by PSCOMP_1:39; p`1 = r by A70,EUCLID:52; then (N-min C)`1 <= G*(i+1,1)`1 by A71,A73,XXREAL_0:2; then A74: N-min C in cell(G,i,width G-'1) by A33,A4,A7,A6,A36,A53,A42,A44,A48,A63 ,A66,A65,A69; N-min C <> G*(i,len G-'1) by A34,A33,A19,A25,A26,A52,A39,SPRECT_3:13; hence thesis by A9,A10,A11,A13,A14,A33,A4,A7,A6,A19,A36,A41,A74,Th29; end; suppose i = i2+1 & len G = j2; then right_cell(f,m,G) = cell(G,i2,len G) & i2 < len G by A1,A45,A5,A6 ,A27,A36,A54,A56,A57,GOBRD13:26,NAT_1:13; hence thesis by A2,A45,A54,JORDAN8:15; end; suppose A75: i = i2 & len G = j2+1; then A76: j2 = len G-'1 by NAT_D:34; then A77: right_cell(f,m,G) = cell(G,i-'1,len G-'1) by A1,A45,A5,A6,A36,A54,A56 ,A57,A75,GOBRD13:28; m-'1 <= m by NAT_D:35; then A78: m-'1 <= len f by A21,XXREAL_0:2; now 1 <= i9+1 by A9,NAT_1:13; then A79: G*(i9+1,len G)`2 = G* (1,len G)`2 by A10,A33,A24,GOBOARD5:1; assume A80: m = 1; G*(i9,len G)`2 = G*(1,len G)`2 by A9,A34,A33,A24,GOBOARD5:1; hence contradiction by A11,A12,A33,A6,A19,A27,A36,A25,A26,A57,A75,A76 ,A80,A79,GOBOARD5:4; end; then m > 1 by A45,XXREAL_0:1; then A81: m-'1 >= 1 by NAT_D:49; A82: m-'1+1 = m by A45,XREAL_1:235; then consider i19,j19,i29,j29 being Element of NAT such that A83: [i19,j19] in Indices G and A84: f/.(m-'1) = G*(i19,j19) and A85: [i29,j29] in Indices G & f/.m = G*(i29,j29) &( i19 = i29 & j19+1 = j29 or i19+1 = i29 & j19 = j29 or i19 = i29+1 & j19 = j29 or i19 = i29 & j19 = j29+1) by A1,A21,A81,JORDAN8:3; A86: 1 <= i19 by A83,MATRIX_1:38; A87: i19 <= len G by A83,MATRIX_1:38; now per cases by A5,A6,A36,A85,GOBOARD1:5; suppose A88: i19 = i & j19+1 = len G; then j19 = len G-'1 by NAT_D:34; then left_cell(f,m-'1,G) = cell(G,i-'1,len G-'1) by A1,A21,A5,A6 ,A36,A81,A82,A83,A84,A88,GOBRD13:21; hence contradiction by A2,A21,A59,A77,A81,A82; end; suppose A89: i19+1 = i & j19 = len G; A90: G*(i19,j)`2 = G*(1,j)`2 & G*(i,j)`2 = G*(1,j)`2 by A19,A27,A22,A28 ,A86,A87,GOBOARD5:1; m-'1 in dom f by A81,A78,FINSEQ_3:25; then A91: f/.(m-'1) in L~f by A30,GOBOARD1:1,XXREAL_0:2; then W-bound L~f <= (f/.(m-'1))`1 & (f/.(m-'1))`1 <= E-bound L~f by PSCOMP_1:24; then f/.(m-'1) in LSeg(NW-corner L~f, NE-corner L~f) by A4,A7,A8,A6 ,A36,A18,A20,A84,A89,A90,GOBOARD7:8; then A92: f/.(m-'1) in LSeg(NW-corner L~f, NE-corner L~f) /\ L~f by A91, XBOOLE_0:def 4; i19 < i by A89,NAT_1:13; then (f/.(m-'1))`1 < (f/.m)`1 by A6,A27,A22,A28,A36,A84,A86,A89, GOBOARD5:3; hence contradiction by A4,A7,A92,PSCOMP_1:39; end; suppose i19 = i+1 & j19 = len G; then right_cell(f,m-'1,G) = cell(G,i,len G) by A1,A21,A5,A6,A36,A81 ,A82,A83,A84,GOBRD13:26; hence contradiction by A2,A21,A27,A81,A82,JORDAN8:15; end; suppose i19 = i & j19 = len G+1; then len G+1 <= len G+0 by A51,A83,MATRIX_1:38; hence contradiction by XREAL_1:6; end; end; hence thesis; end; end; hence thesis; end; end; definition let C be compact non vertical non horizontal non empty Subset of TOP-REAL 2; let n be Nat; assume A1: C is connected; func Cage(C,n) -> clockwise_oriented standard non constant special_circular_sequence means :Def1: it is_sequence_on Gauge(C,n) & (ex i st 1 <= i & i+1 <= len Gauge(C,n) & it/.1 = Gauge(C,n)*(i,width Gauge(C,n)) & it/. 2 = Gauge(C,n)*(i+1,width Gauge(C,n)) & N-min C in cell(Gauge(C,n),i,width Gauge(C,n)-'1) & N-min C <> Gauge(C,n)*(i,width Gauge(C,n)-'1)) & for k st 1 <= k & k+2 <= len it holds (front_left_cell(it,k,Gauge(C,n)) misses C & front_right_cell(it,k,Gauge(C,n)) misses C implies it turns_right k,Gauge(C,n)) & (front_left_cell(it,k,Gauge(C,n)) misses C & front_right_cell(it,k,Gauge(C,n) ) meets C implies it goes_straight k,Gauge(C,n)) & (front_left_cell(it,k,Gauge( C,n)) meets C implies it turns_left k,Gauge(C,n)); existence proof set W = W-bound C, E = E-bound C, S = S-bound C, N = N-bound C; set G = Gauge(C,n); defpred P[Element of NAT,set,set] means ($1 = 0 implies ex i st 1 <= i & i +1 <= len G & N-min C in cell(G,i,width G-'1) & N-min C <> G*(i,width G-'1) & $3 = <*G*(i,width G)*>) & ($1 = 1 implies ex i st 1 <= i & i+1 <= len G & N-min C in cell(G,i,width G-'1) & N-min C <> G*(i,width G-'1) & $3 = <*G*(i,width G), G*(i+1,width G)*>) & ($1 > 1 & $2 is FinSequence of TOP-REAL 2 implies for f being FinSequence of TOP-REAL 2 st $2 = f holds (len f = $1 implies (f is_sequence_on G & right_cell(f,len f-'1,G) meets C implies (front_left_cell(f, (len f)-'1,G) misses C & front_right_cell(f,(len f)-'1,G) misses C implies ex i ,j st f^<*G*(i,j)*> turns_right (len f)-'1,G & $3 = f^<*G*(i,j)*>) & ( front_left_cell(f,(len f)-'1,G) misses C & front_right_cell(f,(len f)-'1,G) meets C implies ex i,j st f^<*G*(i,j)*> goes_straight (len f)-'1,G & $3 = f^<*G *(i,j)*>) & (front_left_cell(f,(len f)-'1,G) meets C implies ex i,j st f^<*G*(i ,j)*> turns_left (len f)-'1,G & $3 = f^<*G*(i,j)*>)) & (not f is_sequence_on G or right_cell(f,len f-'1,G) misses C implies $3 = f^<*G*(1,1)*>)) & (len f <> $1 implies $3 = {})) & ($1 > 1 & $2 is not FinSequence of TOP-REAL 2 implies $3 = {}); A2: len G = width G by JORDAN8:def 1; A3: for k being Element of NAT, x being set ex y being set st P[k,x,y] proof let k be Element of NAT, x be set; consider m being Element of NAT such that A4: 1 <= m & m+1 <= len G & N-min C in cell(G,m,width G-'1) & N-min C <> G*(m,width G-'1) by Th28; per cases by NAT_1:25; suppose A5: k=0; take <*G*(m,width G)*>; thus thesis by A4,A5; end; suppose A6: k = 1; take <*G*(m,width G),G*(m+1,width G)*>; thus thesis by A4,A6; end; suppose that A7: k > 1 and A8: x is FinSequence of TOP-REAL 2; reconsider f = x as FinSequence of TOP-REAL 2 by A8; thus thesis proof per cases; suppose A9: len f = k; thus thesis proof per cases; suppose A10: f is_sequence_on G & right_cell(f,len f-'1,G) meets C; A11: (len f) -'1 +1 = len f by A7,A9,XREAL_1:235; then A12: (len f)-'1+(1+1) = (len f)+1; A13: (len f)-'1+1 in dom f by A7,A9,A11,FINSEQ_3:25; A14: 1 <= (len f)-'1 by A7,A9,NAT_D:49; then consider i1,j1,i2,j2 being Element of NAT such that A15: [i1,j1] in Indices G and A16: f/.((len f) -'1) = G*(i1,j1) and A17: [i2,j2] in Indices G and A18: f/.((len f) -'1+1) = G*(i2,j2) and A19: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2 +1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A10,A11,JORDAN8:3; A20: i1 <= len G by A15,MATRIX_1:38; A21: 1 <= j2+1 by NAT_1:12; A22: 1 <= i2 by A17,MATRIX_1:38; A23: j1 <= width G by A15,MATRIX_1:38; A24: 1 <= i2+1 by NAT_1:12; A25: 1 <= j2 by A17,MATRIX_1:38; (len f)-'1 <= len f by NAT_D:35; then A26: (len f)-'1 in dom f by A14,FINSEQ_3:25; A27: j2 <= width G by A17,MATRIX_1:38; then A28: j2-'1 <= width G by NAT_D:44; A29: i2 <= len G by A17,MATRIX_1:38; then A30: i2-'1 <= len G by NAT_D:44; thus thesis proof per cases; suppose A31: front_left_cell(f,(len f)-'1,G) misses C & front_right_cell(f,(len f)-'1,G) misses C; thus thesis proof per cases by A19; suppose A32: i1 = i2 & j1+1 = j2; take f1 = f^<*G*(i2+1,j2)*>; now take i=i2+1 ,j=j2; thus f1 turns_right (len f)-'1,G proof let i19,j19,i29,j29 be Element of NAT; assume that A33: [i19,j19] in Indices G and A34: [i29,j29] in Indices G and A35: f1/.((len f)-'1) = G*(i19,j19) and A36: f1/.((len f)-'1+1) = G*(i29,j29); A37: f/.((len f)-'1) = G*(i19,j19) by A26,A35, FINSEQ_4:68; then A38: i1 = i19 by A15,A16,A33,GOBOARD1:5; A39: j1 = j19 by A15,A16,A33,A37,GOBOARD1:5; A40: f/.((len f)-'1+1) = G*(i29,j29) by A13,A36, FINSEQ_4:68; then A41: i2 = i29 by A17,A18,A34,GOBOARD1:5; A42: j2 = j29 by A17,A18,A34,A40,GOBOARD1:5; per cases by A15,A16,A19,A33,A37,A41,A42,GOBOARD1:5 ; case i19 = i29 & j19+1 = j29; now assume i2+1 > len G; then A43: (len G)+1 <= i2+1 by NAT_1:13; i2+1 <= (len G)+1 by A29,XREAL_1:6; then i2+1 = (len G)+1 by A43,XXREAL_0:1; then cell(G,len G,j1) meets C by A10,A14,A11 ,A15,A16,A17,A18,A32,GOBRD13:22; hence contradiction by A2,A23,JORDAN8:16; end; hence [i29+1,j29] in Indices G by A25,A27,A24,A41,A42, MATRIX_1:36; thus thesis by A12,A41,A42,FINSEQ_4:67; end; case i19+1 = i29 & j19 = j29; hence thesis by A32,A38,A41; end; case i19 = i29+1 & j19 = j29; hence thesis by A32,A38,A41; end; case i19 = i29 & j19 = j29+1; hence thesis by A32,A39,A42; end; end; end; hence thesis by A7,A9,A10,A31; end; suppose A44: i1+1 = i2 & j1 = j2; take f1 = f^<*G*(i2,j2-'1)*>; now take i=i2,j=j2-'1; thus f1 turns_right (len f)-'1,G proof let i19,j19,i29,j29 be Element of NAT; assume that A45: [i19,j19] in Indices G and A46: [i29,j29] in Indices G and A47: f1/.((len f)-'1) = G*(i19,j19) and A48: f1/.((len f)-'1+1) = G*(i29,j29); A49: f/.((len f)-'1+1) = G*(i29,j29) by A13,A48, FINSEQ_4:68; then A50: i2 = i29 by A17,A18,A46,GOBOARD1:5; A51: f/.((len f)-'1) = G*(i19,j19) by A26,A47, FINSEQ_4:68; then A52: i1 = i19 by A15,A16,A45,GOBOARD1:5; A53: j2 = j29 by A17,A18,A46,A49,GOBOARD1:5; per cases by A15,A16,A19,A45,A51,A50,A53,GOBOARD1:5 ; case i19 = i29 & j19+1 = j29; hence thesis by A44,A52,A50; end; case i19+1 = i29 & j19 = j29; now assume j2-'1 < 1; then j2 <= 1 by NAT_1:14,NAT_D:36; then j2 = 1 by A25,XXREAL_0:1; then cell(G,i1,1-'1) meets C by A10,A14,A11,A15 ,A16,A17,A18,A44,GOBRD13:24; then cell(G,i1,0) meets C by XREAL_1:232; hence contradiction by A20,JORDAN8:17; end; hence [i29,j29-'1] in Indices G by A22,A29,A28 ,A50,A53,MATRIX_1:36; thus thesis by A12,A50,A53,FINSEQ_4:67; end; case i19 = i29+1 & j19 = j29; hence thesis by A44,A52,A50; end; case i19 = i29 & j19 = j29+1; hence thesis by A44,A52,A50; end; end; end; hence thesis by A7,A9,A10,A31; end; suppose A54: i1 = i2+1 & j1 = j2; take f1 = f^<*G*(i2,j2+1)*>; now take i=i2,j=j2+1; thus f1 turns_right (len f)-'1,G proof let i19,j19,i29,j29 be Element of NAT; assume that A55: [i19,j19] in Indices G and A56: [i29,j29] in Indices G and A57: f1/.((len f)-'1) = G*(i19,j19) and A58: f1/.((len f)-'1+1) = G*(i29,j29); A59: f/.((len f)-'1+1) = G*(i29,j29) by A13,A58, FINSEQ_4:68; then A60: i2 = i29 by A17,A18,A56,GOBOARD1:5; A61: f/.((len f)-'1) = G*(i19,j19) by A26,A57, FINSEQ_4:68; then A62: i1 = i19 by A15,A16,A55,GOBOARD1:5; A63: j2 = j29 by A17,A18,A56,A59,GOBOARD1:5; per cases by A15,A16,A19,A55,A61,A60,A63,GOBOARD1:5 ; case i19 = i29 & j19+1 = j29; hence thesis by A54,A62,A60; end; case i19+1 = i29 & j19 = j29; hence thesis by A54,A62,A60; end; case i19 = i29+1 & j19 = j29; now assume j2+1 > len G; then A64: (len G)+1 <= j2+1 by NAT_1:13; j2+1 <= (len G)+1 by A2,A27,XREAL_1:6; then j2+1 = (len G)+1 by A64,XXREAL_0:1; then cell(G,i2,len G) meets C by A10,A14,A11 ,A15,A16,A17,A18,A54,GOBRD13:26; hence contradiction by A29,JORDAN8:15; end; hence [i29,j29+1] in Indices G by A2,A22,A29,A21,A60 ,A63,MATRIX_1:36; thus thesis by A12,A60,A63,FINSEQ_4:67; end; case i19 = i29 & j19 = j29+1; hence thesis by A54,A62,A60; end; end; end; hence thesis by A7,A9,A10,A31; end; suppose A65: i1 = i2 & j1 = j2+1; take f1 = f^<*G*(i2-'1,j2)*>; now take i=i2-'1 ,j=j2; thus f1 turns_right (len f)-'1,G proof let i19,j19,i29,j29 be Element of NAT; assume that A66: [i19,j19] in Indices G and A67: [i29,j29] in Indices G and A68: f1/.((len f)-'1) = G*(i19,j19) and A69: f1/.((len f)-'1+1) = G*(i29,j29); A70: f/.((len f)-'1) = G*(i19,j19) by A26,A68, FINSEQ_4:68; then A71: i1 = i19 by A15,A16,A66,GOBOARD1:5; A72: j1 = j19 by A15,A16,A66,A70,GOBOARD1:5; A73: f/.((len f)-'1+1) = G*(i29,j29) by A13,A69, FINSEQ_4:68; then A74: i2 = i29 by A17,A18,A67,GOBOARD1:5; A75: j2 = j29 by A17,A18,A67,A73,GOBOARD1:5; per cases by A15,A16,A19,A66,A70,A74,A75,GOBOARD1:5 ; case i19 = i29 & j19+1 = j29; hence thesis by A65,A72,A75; end; case i19+1 = i29 & j19 = j29; hence thesis by A65,A71,A74; end; case i19 = i29+1 & j19 = j29; hence thesis by A65,A71,A74; end; case i19 = i29 & j19 = j29+1; now assume i2-'1 < 1; then i2 <= 1 by NAT_1:14,NAT_D:36; then i2 = 1 by A22,XXREAL_0:1; then cell(G,1-'1,j2) meets C by A10,A14,A11,A15 ,A16,A17,A18,A65,GOBRD13:28; then cell(G,0,j2) meets C by XREAL_1:232; hence contradiction by A2,A27,JORDAN8:18; end; hence [i29-'1,j29] in Indices G by A25,A27,A30 ,A74,A75,MATRIX_1:36; thus thesis by A12,A74,A75,FINSEQ_4:67; end; end; end; hence thesis by A7,A9,A10,A31; end; end; end; suppose A76: front_left_cell(f,(len f)-'1,G) misses C & front_right_cell(f,(len f)-'1,G) meets C; thus thesis proof per cases by A19; suppose A77: i1 = i2 & j1+1 = j2; take f1 = f^<*G*(i2,j2+1)*>; now take i=i2,j=j2+1; thus f1 goes_straight (len f)-'1,G proof let i19,j19,i29,j29 be Element of NAT; assume that A78: [i19,j19] in Indices G and A79: [i29,j29] in Indices G and A80: f1/.((len f)-'1) = G*(i19,j19) and A81: f1/.((len f)-'1+1) = G*(i29,j29); A82: f/.((len f)-'1) = G*(i19,j19) by A26,A80, FINSEQ_4:68; then A83: i1 = i19 by A15,A16,A78,GOBOARD1:5; A84: j1 = j19 by A15,A16,A78,A82,GOBOARD1:5; A85: f/.((len f)-'1+1) = G*(i29,j29) by A13,A81, FINSEQ_4:68; then A86: i2 = i29 by A17,A18,A79,GOBOARD1:5; A87: j2 = j29 by A17,A18,A79,A85,GOBOARD1:5; per cases by A15,A16,A19,A78,A82,A86,A87,GOBOARD1:5 ; case i19 = i29 & j19+1 = j29; now assume j2+1 > len G; then A88: (len G)+1 <= j2+1 by NAT_1:13; j2+1 <= (len G)+1 by A2,A27,XREAL_1:6; then j2+1 = (len G)+1 by A88,XXREAL_0:1; then cell(G,i1,len G) meets C by A10,A14,A11 ,A15,A16,A17,A18,A76,A77,GOBRD13:35; hence contradiction by A20,JORDAN8:15; end; hence [i29,j29+1] in Indices G by A2,A22,A29,A21,A86 ,A87,MATRIX_1:36; thus thesis by A12,A86,A87,FINSEQ_4:67; end; case i19+1 = i29 & j19 = j29; hence thesis by A77,A83,A86; end; case i19 = i29+1 & j19 = j29; hence thesis by A77,A83,A86; end; case i19 = i29 & j19 = j29+1; hence thesis by A77,A84,A87; end; end; end; hence thesis by A7,A9,A10,A76; end; suppose A89: i1+1 = i2 & j1 = j2; take f1 = f^<*G*(i2+1,j2)*>; now take i=i2+1 ,j=j2; thus f1 goes_straight (len f)-'1,G proof let i19,j19,i29,j29 be Element of NAT; assume that A90: [i19,j19] in Indices G and A91: [i29,j29] in Indices G and A92: f1/.((len f)-'1) = G*(i19,j19) and A93: f1/.((len f)-'1+1) = G*(i29,j29); A94: f/.((len f)-'1+1) = G*(i29,j29) by A13,A93, FINSEQ_4:68; then A95: i2 = i29 by A17,A18,A91,GOBOARD1:5; A96: f/.((len f)-'1) = G*(i19,j19) by A26,A92, FINSEQ_4:68; then A97: i1 = i19 by A15,A16,A90,GOBOARD1:5; A98: j2 = j29 by A17,A18,A91,A94,GOBOARD1:5; per cases by A15,A16,A19,A90,A96,A95,A98,GOBOARD1:5 ; case i19 = i29 & j19+1 = j29; hence thesis by A89,A97,A95; end; case i19+1 = i29 & j19 = j29; now assume i2+1 > len G; then A99: (len G)+1 <= i2+1 by NAT_1:13; i2+1 <= (len G)+1 by A29,XREAL_1:6; then i2+1 = (len G)+1 by A99,XXREAL_0:1; then cell(G,len G,j1-'1) meets C by A10,A14,A11 ,A15,A16,A17,A18,A76,A89,GOBRD13:37; hence contradiction by A2,A23,JORDAN8:16 ,NAT_D:44; end; hence [i29+1,j29] in Indices G by A25,A27,A24,A95 ,A98,MATRIX_1:36; thus thesis by A12,A95,A98,FINSEQ_4:67; end; case i19 = i29+1 & j19 = j29; hence thesis by A89,A97,A95; end; case i19 = i29 & j19 = j29+1; hence thesis by A89,A97,A95; end; end; end; hence thesis by A7,A9,A10,A76; end; suppose A100: i1 = i2+1 & j1 = j2; take f1 = f^<*G*(i2-'1,j2)*>; now take i=i2-'1 ,j=j2; thus f1 goes_straight (len f)-'1,G proof let i19,j19,i29,j29 be Element of NAT; assume that A101: [i19,j19] in Indices G and A102: [i29,j29] in Indices G and A103: f1/.((len f)-'1) = G*(i19,j19) and A104: f1/.((len f)-'1+1) = G*(i29,j29); A105: f/.((len f)-'1+1) = G*(i29,j29) by A13,A104, FINSEQ_4:68; then A106: i2 = i29 by A17,A18,A102,GOBOARD1:5; A107: f/.((len f)-'1) = G*(i19,j19) by A26,A103, FINSEQ_4:68; then A108: i1 = i19 by A15,A16,A101,GOBOARD1:5; A109: j2 = j29 by A17,A18,A102,A105,GOBOARD1:5; per cases by A15,A16,A19,A101,A107,A106,A109, GOBOARD1:5; case i19 = i29 & j19+1 = j29; hence thesis by A100,A108,A106; end; case i19+1 = i29 & j19 = j29; hence thesis by A100,A108,A106; end; case i19 = i29+1 & j19 = j29; now assume i2-'1 < 1; then i2 <= 1 by NAT_1:14,NAT_D:36; then i2 = 1 by A22,XXREAL_0:1; then cell(G,1-'1,j1) meets C by A10,A14,A11,A15 ,A16,A17,A18,A76,A100,GOBRD13:39; then cell(G,0,j1) meets C by XREAL_1:232; hence contradiction by A2,A23,JORDAN8:18; end; hence [i29-'1,j29] in Indices G by A25,A27,A30 ,A106,A109,MATRIX_1:36; thus thesis by A12,A106,A109,FINSEQ_4:67; end; case i19 = i29 & j19 = j29+1; hence thesis by A100,A108,A106; end; end; end; hence thesis by A7,A9,A10,A76; end; suppose A110: i1 = i2 & j1 = j2+1; take f1 = f^<*G*(i2,j2-'1)*>; now take i=i2,j=j2-'1; thus f1 goes_straight (len f)-'1,G proof let i19,j19,i29,j29 be Element of NAT; assume that A111: [i19,j19] in Indices G and A112: [i29,j29] in Indices G and A113: f1/.((len f)-'1) = G*(i19,j19) and A114: f1/.((len f)-'1+1) = G*(i29,j29); A115: f/.((len f)-'1) = G*(i19,j19) by A26,A113, FINSEQ_4:68; then A116: i1 = i19 by A15,A16,A111,GOBOARD1:5; A117: j1 = j19 by A15,A16,A111,A115,GOBOARD1:5; A118: f/.((len f)-'1+1) = G*(i29,j29) by A13,A114, FINSEQ_4:68; then A119: i2 = i29 by A17,A18,A112,GOBOARD1:5; A120: j2 = j29 by A17,A18,A112,A118,GOBOARD1:5; per cases by A15,A16,A19,A111,A115,A119,A120, GOBOARD1:5; case i19 = i29 & j19+1 = j29; hence thesis by A110,A117,A120; end; case i19+1 = i29 & j19 = j29; hence thesis by A110,A116,A119; end; case i19 = i29+1 & j19 = j29; hence thesis by A110,A116,A119; end; case i19 = i29 & j19 = j29+1; now assume j2-'1 < 1; then j2 <= 1 by NAT_1:14,NAT_D:36; then j2 = 1 by A25,XXREAL_0:1; then cell(G,i1-'1,1-'1) meets C by A10,A14,A11 ,A15,A16,A17,A18,A76,A110,GOBRD13:41; then cell(G,i1-'1,0) meets C by XREAL_1:232; hence contradiction by A20,JORDAN8:17,NAT_D:44; end; hence [i29,j29-'1] in Indices G by A22,A29,A28 ,A119,A120,MATRIX_1:36; thus thesis by A12,A119,A120,FINSEQ_4:67; end; end; end; hence thesis by A7,A9,A10,A76; end; end; end; suppose A121: front_left_cell(f,(len f)-'1,G) meets C; thus thesis proof per cases by A19; suppose A122: i1 = i2 & j1+1 = j2; take f1 = f^<*G*(i2-'1,j2)*>; now take i=i2-'1 ,j=j2; thus f1 turns_left (len f)-'1,G proof let i19,j19,i29,j29 be Element of NAT; assume that A123: [i19,j19] in Indices G and A124: [i29,j29] in Indices G and A125: f1/.((len f)-'1) = G*(i19,j19) and A126: f1/.((len f)-'1+1) = G*(i29,j29); A127: f/.((len f)-'1) = G*(i19,j19) by A26,A125, FINSEQ_4:68; then A128: i1 = i19 by A15,A16,A123,GOBOARD1:5; A129: j1 = j19 by A15,A16,A123,A127,GOBOARD1:5; A130: f/.((len f)-'1+1) = G*(i29,j29) by A13,A126, FINSEQ_4:68; then A131: i2 = i29 by A17,A18,A124,GOBOARD1:5; A132: j2 = j29 by A17,A18,A124,A130,GOBOARD1:5; per cases by A15,A16,A19,A123,A127,A131,A132, GOBOARD1:5; case i19 = i29 & j19+1 = j29; now assume i2-'1 < 1; then i2 <= 1 by NAT_1:14,NAT_D:36; then i2 = 1 by A22,XXREAL_0:1; then cell(G,1-'1,j2) meets C by A10,A14,A11,A15 ,A16,A17,A18,A121,A122,GOBRD13:34; then cell(G,0,j2) meets C by XREAL_1:232; hence contradiction by A2,A27,JORDAN8:18; end; hence [i29-'1,j29] in Indices G by A25,A27,A30 ,A131,A132,MATRIX_1:36; thus thesis by A12,A131,A132,FINSEQ_4:67; end; case i19+1 = i29 & j19 = j29; hence thesis by A122,A128,A131; end; case i19 = i29+1 & j19 = j29; hence thesis by A122,A128,A131; end; case i19 = i29 & j19 = j29+1; hence thesis by A122,A129,A132; end; end; end; hence thesis by A7,A9,A10,A121; end; suppose A133: i1+1 = i2 & j1 = j2; take f1 = f^<*G*(i2,j2+1)*>; now take i=i2,j=j2+1; thus f1 turns_left (len f)-'1,G proof let i19,j19,i29,j29 be Element of NAT; assume that A134: [i19,j19] in Indices G and A135: [i29,j29] in Indices G and A136: f1/.((len f)-'1) = G*(i19,j19) and A137: f1/.((len f)-'1+1) = G*(i29,j29); A138: f/.((len f)-'1+1) = G*(i29,j29) by A13,A137, FINSEQ_4:68; then A139: i2 = i29 by A17,A18,A135,GOBOARD1:5; A140: f/.((len f)-'1) = G*(i19,j19) by A26,A136, FINSEQ_4:68; then A141: i1 = i19 by A15,A16,A134,GOBOARD1:5; A142: j2 = j29 by A17,A18,A135,A138,GOBOARD1:5; per cases by A15,A16,A19,A134,A140,A139,A142, GOBOARD1:5; case i19 = i29 & j19+1 = j29; hence thesis by A133,A141,A139; end; case i19+1 = i29 & j19 = j29; now assume j2+1 > len G; then A143: (len G)+1 <= j2+1 by NAT_1:13; j2+1 <= (len G)+1 by A2,A27,XREAL_1:6; then j2+1 = (len G)+1 by A143,XXREAL_0:1; then cell(G,i2,len G) meets C by A10,A14,A11 ,A15,A16,A17,A18,A121,A133,GOBRD13:36; hence contradiction by A29,JORDAN8:15; end; hence [i29,j29+1] in Indices G by A2,A22,A29,A21,A139 ,A142,MATRIX_1:36; thus thesis by A12,A139,A142,FINSEQ_4:67; end; case i19 = i29+1 & j19 = j29; hence thesis by A133,A141,A139; end; case i19 = i29 & j19 = j29+1; hence thesis by A133,A141,A139; end; end; end; hence thesis by A7,A9,A10,A121; end; suppose A144: i1 = i2+1 & j1 = j2; take f1 = f^<*G*(i2,j2-'1)*>; now take i=i2,j=j2-'1; thus f1 turns_left (len f)-'1,G proof let i19,j19,i29,j29 be Element of NAT; assume that A145: [i19,j19] in Indices G and A146: [i29,j29] in Indices G and A147: f1/.((len f)-'1) = G*(i19,j19) and A148: f1/.((len f)-'1+1) = G*(i29,j29); A149: f/.((len f)-'1+1) = G*(i29,j29) by A13,A148, FINSEQ_4:68; then A150: i2 = i29 by A17,A18,A146,GOBOARD1:5; A151: f/.((len f)-'1) = G*(i19,j19) by A26,A147, FINSEQ_4:68; then A152: i1 = i19 by A15,A16,A145,GOBOARD1:5; A153: j2 = j29 by A17,A18,A146,A149,GOBOARD1:5; per cases by A15,A16,A19,A145,A151,A150,A153, GOBOARD1:5; case i19 = i29 & j19+1 = j29; hence thesis by A144,A152,A150; end; case i19+1 = i29 & j19 = j29; hence thesis by A144,A152,A150; end; case i19 = i29+1 & j19 = j29; now assume j2-'1 < 1; then j2 <= 1 by NAT_1:14,NAT_D:36; then j2 = 1 by A25,XXREAL_0:1; then cell(G,i2-'1,1-'1) meets C by A10,A14,A11 ,A15,A16,A17,A18,A121,A144,GOBRD13:38; then cell(G,i2-'1,0) meets C by XREAL_1:232; hence contradiction by A29,JORDAN8:17,NAT_D:44; end; hence [i29,j29-'1] in Indices G by A22,A29,A28 ,A150,A153,MATRIX_1:36; thus thesis by A12,A150,A153,FINSEQ_4:67; end; case i19 = i29 & j19 = j29+1; hence thesis by A144,A152,A150; end; end; end; hence thesis by A7,A9,A10,A121; end; suppose A154: i1 = i2 & j1 = j2+1; take f1 = f^<*G*(i2+1,j2)*>; now take i=i2+1 ,j=j2; thus f1 turns_left (len f)-'1,G proof let i19,j19,i29,j29 be Element of NAT; assume that A155: [i19,j19] in Indices G and A156: [i29,j29] in Indices G and A157: f1/.((len f)-'1) = G*(i19,j19) and A158: f1/.((len f)-'1+1) = G*(i29,j29); A159: f/.((len f)-'1) = G*(i19,j19) by A26,A157, FINSEQ_4:68; then A160: i1 = i19 by A15,A16,A155,GOBOARD1:5; A161: j1 = j19 by A15,A16,A155,A159,GOBOARD1:5; A162: f/.((len f)-'1+1) = G*(i29,j29) by A13,A158, FINSEQ_4:68; then A163: i2 = i29 by A17,A18,A156,GOBOARD1:5; A164: j2 = j29 by A17,A18,A156,A162,GOBOARD1:5; per cases by A15,A16,A19,A155,A159,A163,A164, GOBOARD1:5; case i19 = i29 & j19+1 = j29; hence thesis by A154,A161,A164; end; case i19+1 = i29 & j19 = j29; hence thesis by A154,A160,A163; end; case i19 = i29+1 & j19 = j29; hence thesis by A154,A160,A163; end; case i19 = i29 & j19 = j29+1; now assume i2+1 > len G; then A165: (len G)+1 <= i2+1 by NAT_1:13; i2+1 <= (len G)+1 by A29,XREAL_1:6; then i2+1 = (len G)+1 by A165,XXREAL_0:1; then cell(G,len G,j2-'1) meets C by A10,A14,A11 ,A15,A16,A17,A18,A121,A154,GOBRD13:40; hence contradiction by A2,A27,JORDAN8:16 ,NAT_D:44; end; hence [i29+1,j29] in Indices G by A25,A27,A24 ,A163,A164,MATRIX_1:36; thus thesis by A12,A163,A164,FINSEQ_4:67; end; end; end; hence thesis by A7,A9,A10,A121; end; end; end; end; end; suppose A166: not f is_sequence_on G or right_cell(f,len f-'1,G) misses C; take f^<*G*(1,1)*>; thus thesis by A7,A9,A166; end; end; end; suppose A167: len f <> k; take {}; thus thesis by A7,A167; end; end; end; suppose A168: k > 1 & x is not FinSequence of TOP-REAL 2; take {}; thus thesis by A168; end; end; consider F being Function such that A169: dom F = NAT and A170: F.0 = {} and A171: for k being Element of NAT holds P[k,F.k,F.(k+1)] from RECDEF_1: sch 1 (A3); defpred P[Element of NAT] means F.$1 is FinSequence of TOP-REAL 2; A172: {} = <*>(the carrier of TOP-REAL 2); A173: for k st P[k] holds P[k+1] proof let k such that A174: F.k is FinSequence of TOP-REAL 2; A175: P[k,F.k,F.(k+1)] by A171; per cases by NAT_1:25; suppose k = 0; hence thesis by A175; end; suppose k = 1; hence thesis by A175; end; suppose A176: k > 1; thus thesis proof reconsider f = F.k as FinSequence of TOP-REAL 2 by A174; per cases; suppose A177: len f = k; thus thesis proof per cases; suppose A178: f is_sequence_on G & right_cell(f,len f-'1,G) meets C; then A179: front_left_cell(f,(len f)-'1,G) meets C implies ex i,j st f^<*G*(i,j) *> turns_left (len f)-'1,G & F.(k+1) = f^<*G*(i,j)*> by A171 ,A176,A177; A180: front_left_cell(f,(len f)-'1,G) misses C & front_right_cell(f,(len f) -'1,G) meets C implies ex i,j st f^<*G*(i,j)*> goes_straight (len f)-'1,G & F.( k+1) = f^<*G*(i,j)*> by A171,A176,A177,A178; front_left_cell(f,(len f)-'1,G) misses C & front_right_cell(f,(len f) -'1,G) misses C implies ex i,j st f^<*G*(i,j)*> turns_right (len f)-'1,G & F.(k +1) = f^<*G*(i,j)*> by A171,A176,A177,A178; hence thesis by A180,A179; end; suppose A181: not f is_sequence_on G or right_cell(f,len f-'1,G) misses C; f^<*G*(1,1)*> is FinSequence of TOP-REAL 2; hence thesis by A171,A176,A177,A181; end; end; end; suppose len f <> k; hence thesis by A171,A172,A176; end; end; end; end; A182: P[0] by A170,A172; A183: for k holds P[k] from NAT_1:sch 1(A182,A173); rng F c= (the carrier of TOP-REAL 2)* proof let y be set; assume y in rng F; then ex x being set st x in dom F & F.x = y by FUNCT_1:def 3; then y is FinSequence of TOP-REAL 2 by A169,A183; hence thesis by FINSEQ_1:def 11; end; then reconsider F as Function of NAT,(the carrier of TOP-REAL 2)* by A169, FUNCT_2:def 1,RELSET_1:4; defpred P[Element of NAT] means len(F.$1) = $1; A184: for k st P[k] holds P[k+1] proof let k such that A185: len(F.k) = k; A186: P[k,F.k,F.(k+1)] by A171; per cases by NAT_1:25; suppose k = 0; hence thesis by A186,FINSEQ_1:39; end; suppose k = 1; hence thesis by A186,FINSEQ_1:44; end; suppose A187: k > 1; thus thesis proof per cases; suppose A188: F.k is_sequence_on G & right_cell(F.k,len(F.k)-'1,G) meets C; then A189: front_left_cell(F.k,(len(F.k))-'1,G) meets C implies ex i,j st (F.k)^ <*G*(i,j)*> turns_left (len(F.k))-'1,G & F.(k+1) = (F.k)^<*G*(i,j)*> by A171,A185,A187; A190: front_left_cell(F.k,(len(F.k))-'1,G) misses C & front_right_cell(F.k, (len(F.k))-'1,G) meets C implies ex i,j st (F.k)^<*G*(i,j )*> goes_straight (len (F.k))-'1,G & F.(k+1) = (F.k)^<*G*(i,j)*> by A171,A185 ,A187,A188; front_left_cell(F.k,(len(F.k))-'1,G) misses C & front_right_cell(F.k, (len(F.k))-'1,G) misses C implies ex i,j st (F.k)^<*G*(i, j)*> turns_right (len( F.k))-'1,G & F.(k+1) = (F.k)^<*G*(i,j)*> by A171,A185 ,A187,A188; hence thesis by A185,A190,A189,FINSEQ_2:16; end; suppose not F.k is_sequence_on G or right_cell(F.k,len(F.k)-'1,G ) misses C; then F.(k+1) = (F.k)^<*G*(1,1)*> by A171,A185,A187; hence thesis by A185,FINSEQ_2:16; end; end; end; end; A191: P[0] by A170,CARD_1:27; A192: for k holds P[k] from NAT_1:sch 1(A191,A184); A193: now let k such that A194: F.k is_sequence_on G and A195: for m st 1 <= m & m+1 <= len(F.k) holds left_cell(F.k,m,G) misses C & right_cell(F.k,m,G) meets C and A196: k > 1; len(F.k) = k by A192; then A197: 1 <= (len(F.k))-'1 & (len(F.k)) -'1 +1 = len(F.k) by A196,NAT_D:49 ,XREAL_1:235; then A198: right_cell(F.k,(len(F.k))-'1,G) meets C by A195; let i1,j1,i2,j2 be Element of NAT such that A199: [i1,j1] in Indices G and A200: (F.k)/.((len(F.k)) -'1) = G*(i1,j1) and A201: [i2,j2] in Indices G and A202: (F.k)/.len(F.k) = G*(i2,j2); A203: i2 <= len G by A201,MATRIX_1:38; A204: 1 <= i2+1 by NAT_1:12; A205: 1 <= j2 by A201,MATRIX_1:38; A206: j2 <= width G by A201,MATRIX_1:38; A207: j1 <= width G by A199,MATRIX_1:38; hereby assume A208: i1 = i2 & j1+1 = j2; now assume i2+1 > len G; then A209: (len G)+1 <= i2+1 by NAT_1:13; i2+1 <= (len G)+1 by A203,XREAL_1:6; then i2+1 = (len G)+1 by A209,XXREAL_0:1; then cell(G,len G,j1) meets C by A194,A199,A200,A201,A202,A197,A198 ,A208,GOBRD13:22; hence contradiction by A2,A207,JORDAN8:16; end; hence [i2+1,j2] in Indices G by A205,A206,A204,MATRIX_1:36; end; A210: i1 <= len G by A199,MATRIX_1:38; A211: 1 <= i2 by A201,MATRIX_1:38; A212: j2-'1 <= width G by A206,NAT_D:44; hereby assume A213: i1+1 = i2 & j1 = j2; now assume j2-'1 < 1; then j2 <= 1 by NAT_1:14,NAT_D:36; then j2 = 1 by A205,XXREAL_0:1; then cell(G,i1,1-'1) meets C by A194,A199,A200,A201,A202,A197,A198 ,A213,GOBRD13:24; then cell(G,i1,0) meets C by XREAL_1:232; hence contradiction by A210,JORDAN8:17; end; hence [i2,j2-'1] in Indices G by A211,A203,A212,MATRIX_1:36; end; A214: 1 <= j2+1 by NAT_1:12; hereby assume A215: i1 = i2+1 & j1 = j2; now assume j2+1 > len G; then A216: (len G)+1 <= j2+1 by NAT_1:13; j2+1 <= (len G)+1 by A2,A206,XREAL_1:6; then j2+1 = (len G)+1 by A216,XXREAL_0:1; then cell(G,i2,len G) meets C by A194,A199,A200,A201,A202,A197,A198 ,A215,GOBRD13:26; hence contradiction by A203,JORDAN8:15; end; hence [i2,j2+1] in Indices G by A2,A211,A203,A214,MATRIX_1:36; end; A217: i2-'1 <= len G by A203,NAT_D:44; hereby assume A218: i1 = i2 & j1 = j2+1; now assume i2-'1 < 1; then i2 <= 1 by NAT_1:14,NAT_D:36; then i2 = 1 by A211,XXREAL_0:1; then cell(G,1-'1,j2) meets C by A194,A199,A200,A201,A202,A197,A198 ,A218,GOBRD13:28; then cell(G,0,j2) meets C by XREAL_1:232; hence contradiction by A2,A206,JORDAN8:18; end; hence [i2-'1,j2] in Indices G by A205,A206,A217,MATRIX_1:36; end; hereby assume A219: front_right_cell(F.k,(len(F.k))-'1,G) meets C & i1 = i2 & j1 +1 = j2; now assume j2+1 > len G; then A220: (len G)+1 <= j2+1 by NAT_1:13; j2+1 <= (len G)+1 by A2,A206,XREAL_1:6; then j2+1 = (len G)+1 by A220,XXREAL_0:1; then cell(G,i1,len G) meets C by A194,A199,A200,A201,A202,A197,A219, GOBRD13:35; hence contradiction by A210,JORDAN8:15; end; hence [i2,j2+1] in Indices G by A2,A211,A203,A214,MATRIX_1:36; end; hereby assume A221: front_right_cell(F.k,(len(F.k))-'1,G) meets C & i1+1 = i2 & j1 = j2; now assume i2+1 > len G; then A222: (len G)+1 <= i2+1 by NAT_1:13; i2+1 <= (len G)+1 by A203,XREAL_1:6; then i2+1 = (len G)+1 by A222,XXREAL_0:1; then cell(G,len G,j1-'1) meets C by A194,A199,A200,A201,A202,A197 ,A221,GOBRD13:37; hence contradiction by A2,A207,JORDAN8:16,NAT_D:44; end; hence [i2+1,j2] in Indices G by A205,A206,A204,MATRIX_1:36; end; hereby assume A223: front_right_cell(F.k,(len(F.k))-'1,G) meets C & i1 = i2+1 & j1 = j2; now assume i2-'1 < 1; then i2 <= 1 by NAT_1:14,NAT_D:36; then i2 = 1 by A211,XXREAL_0:1; then cell(G,1-'1,j1) meets C by A194,A199,A200,A201,A202,A197,A223, GOBRD13:39; then cell(G,0,j1) meets C by XREAL_1:232; hence contradiction by A2,A207,JORDAN8:18; end; hence [i2-'1,j2] in Indices G by A205,A206,A217,MATRIX_1:36; end; hereby assume A224: front_right_cell(F.k,(len(F.k))-'1,G) meets C & i1 = i2 & j1 = j2+1; now assume j2-'1 < 1; then j2 <= 1 by NAT_1:14,NAT_D:36; then j2 = 1 by A205,XXREAL_0:1; then cell(G,i1-'1,1-'1) meets C by A194,A199,A200,A201,A202,A197,A224 ,GOBRD13:41; then cell(G,i1-'1,0) meets C by XREAL_1:232; hence contradiction by A210,JORDAN8:17,NAT_D:44; end; hence [i2,j2-'1] in Indices G by A211,A203,A212,MATRIX_1:36; end; hereby assume A225: front_left_cell(F.k,(len(F.k))-'1,G) meets C & i1 = i2 & j1+ 1 = j2; now assume i2-'1 < 1; then i2 <= 1 by NAT_1:14,NAT_D:36; then i2 = 1 by A211,XXREAL_0:1; then cell(G,1-'1,j2) meets C by A194,A199,A200,A201,A202,A197,A225, GOBRD13:34; then cell(G,0,j2) meets C by XREAL_1:232; hence contradiction by A2,A206,JORDAN8:18; end; hence [i2-'1,j2] in Indices G by A205,A206,A217,MATRIX_1:36; end; hereby assume A226: front_left_cell(F.k,(len(F.k))-'1,G) meets C & i1+1 = i2 & j1 = j2; now assume j2+1 > len G; then A227: (len G)+1 <= j2+1 by NAT_1:13; j2+1 <= (len G)+1 by A2,A206,XREAL_1:6; then j2+1 = (len G)+1 by A227,XXREAL_0:1; then cell(G,i2,len G) meets C by A194,A199,A200,A201,A202,A197,A226, GOBRD13:36; hence contradiction by A203,JORDAN8:15; end; hence [i2,j2+1] in Indices G by A2,A211,A203,A214,MATRIX_1:36; end; hereby assume A228: front_left_cell(F.k,(len(F.k))-'1,G) meets C & i1 = i2+1 & j1 = j2; now assume j2-'1 < 1; then j2 <= 1 by NAT_1:14,NAT_D:36; then j2 = 1 by A205,XXREAL_0:1; then cell(G,i2-'1,1-'1) meets C by A194,A199,A200,A201,A202,A197,A228 ,GOBRD13:38; then cell(G,i2-'1,0) meets C by XREAL_1:232; hence contradiction by A203,JORDAN8:17,NAT_D:44; end; hence [i2,j2-'1] in Indices G by A211,A203,A212,MATRIX_1:36; end; hereby assume A229: front_left_cell(F.k,(len(F.k))-'1,G) meets C & i1 = i2 & j1 = j2+1; now assume i2+1 > len G; then A230: (len G)+1 <= i2+1 by NAT_1:13; i2+1 <= (len G)+1 by A203,XREAL_1:6; then i2+1 = (len G)+1 by A230,XXREAL_0:1; then cell(G,len G,j2-'1) meets C by A194,A199,A200,A201,A202,A197 ,A229,GOBRD13:40; hence contradiction by A2,A206,JORDAN8:16,NAT_D:44; end; hence [i2+1,j2] in Indices G by A205,A206,A204,MATRIX_1:36; end; end; defpred P[Element of NAT] means F.$1 is_sequence_on G & for m st 1 <= m & m+1 <= len(F.$1) holds left_cell(F.$1,m,G) misses C & right_cell(F.$1,m,G) meets C; A231: len G = 2|^n+3 by JORDAN8:def 1; A232: for k st P[k] holds P[k+1] proof A233: 2|^n > 0 by NEWTON:83; A234: 1 <= len G by A231,NAT_1:12; let k such that A235: F.k is_sequence_on G and A236: for m st 1 <= m & m+1 <= len(F.k) holds left_cell(F.k,m,G) misses C & right_cell(F.k,m,G) meets C; A237: len(F.k) = k by A192; A238: len(F.(k+1)) = k+1 by A192; per cases by NAT_1:25; suppose A239: k = 0; then consider i such that A240: 1 <= i and A241: i+1 <= len G and N-min C in cell(G,i,width G-'1) and N-min C <> G*(i,width G-'1) and A242: F.(k+1) = <*G*(i,width G)*> by A171; i < len G by A241,NAT_1:13; then A243: [i,len G] in Indices G by A2,A234,A240,MATRIX_1:36; A244: now let l; assume l in dom(F.(k+1)); then 1 <= l & l <= 1 by A238,A239,FINSEQ_3:25; then l = 1 by XXREAL_0:1; hence ex i,j st [i,j] in Indices G & (F.(k+1))/.l = G* (i,j) by A2 ,A242,A243,FINSEQ_4:16; end; now let l; assume that A245: l in dom(F.(k+1)) and A246: l+1 in dom(F.(k+1)); 1 <= l & l <= 1 by A238,A239,A245,FINSEQ_3:25; then l = 1 by XXREAL_0:1; hence for i1,j1,i2,j2 st [i1,j1] in Indices G & [i2,j2] in Indices G & (F.(k+1))/.l = G*(i1,j1) & (F.(k+1))/.(l+1) = G*(i2,j2) holds abs(i1-i2)+abs( j1-j2) = 1 by A238,A239,A246,FINSEQ_3:25; end; hence F.(k+1) is_sequence_on G by A244,GOBOARD1:def 9; let m; assume that A247: 1 <= m and A248: m+1 <= len(F.(k+1)); 1 <= m+1 by NAT_1:12; then m+1 = 0+1 by A238,A239,A248,XXREAL_0:1; hence thesis by A247; end; suppose A249: k = 1; then consider i such that A250: 1 <= i and A251: i+1 <= len G and A252: N-min C in cell(G,i,width G-'1) and N-min C <> G*(i,width G-'1) and A253: F.(k+1) = <*G*(i,width G),G*(i+1,width G)*> by A171; A254: i < len G by A251,NAT_1:13; then A255: [i,len G] in Indices G by A2,A234,A250,MATRIX_1:36; 1 <= i+1 by A250,NAT_1:13; then A256: [i+1,len G] in Indices G by A2,A234,A251,MATRIX_1:36; A257: (F.(k+1))/.1 = G*(i,width G) & (F.(k+1))/.2 = G*(i+1,width G) by A253, FINSEQ_4:17; A258: now let l; assume that A259: l in dom(F.(k+1)) and A260: l+1 in dom(F.(k+1)); l <= 2 by A238,A249,A259,FINSEQ_3:25; then A261: l = 0 or l = 1 or l = 2 by NAT_1:26; let i1,j1,i2,j2 such that A262: [i1,j1] in Indices G & [i2,j2] in Indices G & (F.(k+1))/.l = G*(i1,j1) & (F.(k+1))/.(l+1) = G*(i2,j2); j1 = len G & j2 = len G by A2,A238,A249,A257,A255,A256,A259,A260,A261 ,A262,FINSEQ_3:25,GOBOARD1:5; then A263: abs(j1-j2) = 0 by ABSVALUE:def 1; i1 = i & i2 = i+1 by A2,A238,A249,A257,A255,A256,A259,A260,A261,A262, FINSEQ_3:25,GOBOARD1:5; then abs(i2-i1) = 1 by ABSVALUE:def 1; hence abs(i1-i2)+abs(j1-j2) = 1 by A263,UNIFORM1:11; end; now let l; assume A264: l in dom(F.(k+1)); then l <= 2 by A238,A249,FINSEQ_3:25; then l = 0 or l = 1 or l = 2 by NAT_1:26; hence ex i,j st [i,j] in Indices G & (F.(k+1))/.l = G*(i,j) by A2 ,A257,A255,A256,A264,FINSEQ_3:25; end; hence A265: F.(k+1) is_sequence_on G by A258,GOBOARD1:def 9; A266: i < i+1 & i+1 < (i+1)+1 by NAT_1:13; let m; assume that A267: 1 <= m and A268: m+1 <= len(F.(k+1)); 1+1 <= m+1 by A267,XREAL_1:6; then A269: m+1 = 1+1 by A238,A249,A268,XXREAL_0:1; then A270: left_cell(F.(k+1),m,G) = cell(G,i,len G) by A2,A257,A255,A256,A265,A268 ,A266,GOBRD13:def 3; now N > S by JORDAN8:9; then N-S > S-S by XREAL_1:9; then (N-S)/(2|^n) > 0 by A233,XREAL_1:139; then A271: N+0 < N+(N-S)/(2|^n) by XREAL_1:6; [1,len G] in Indices G by A2,A234,MATRIX_1:36; then G* (1,len G) = |[W+((E-W)/(2|^n))*(1-2),S+((N-S)/(2|^n))*((len G)-2)]| by JORDAN8:def 1; then A272: G*(1,len G)`2 = S+((N-S)/(2|^n))*((len G)-2) by EUCLID:52; A273: cell(G,i,len G) = { |[r,s]|: G*(i,1)`1 <= r & r <= G*(i+1,1)`1 & G*(1,len G)`2 <= s } by A2,A250,A254,GOBRD11:31; assume left_cell(F.(k+1),m,G) meets C; then consider p being set such that A274: p in cell(G,i,len G) and A275: p in C by A270,XBOOLE_0:3; reconsider p as Point of TOP-REAL 2 by A274; reconsider p as Element of TOP-REAL 2; A276: p`2 <= N by A275,PSCOMP_1:24; consider r,s such that A277: p = |[r,s]| and G*(i,1)`1 <= r and r <= G*(i+1,1)`1 and A278: G*(1,len G)`2 <= s by A274,A273; ((N-S)/(2|^n))*((len G)-2) = ((N-S)/(2|^n))*(2|^n)+((N-S)/(2|^ n))*1 by A231 .= (N-S)+(N-S)/(2|^n) by A233,XCMPLX_1:87; then N < s by A278,A272,A271,XXREAL_0:2; hence contradiction by A277,A276,EUCLID:52; end; hence left_cell(F.(k+1),m,G) misses C; N-min C in C & N-min C in right_cell(F.(k+1),m,G) by A2,A252,A257,A255 ,A256,A265,A268,A269,A266,GOBRD13:def 2,SPRECT_1:11; hence thesis by XBOOLE_0:3; end; suppose A279: k > 1; then A280: len(F.k) in dom(F.k) by A237,FINSEQ_3:25; A281: (len(F.k)) -'1 +1 = len(F.k) by A237,A279,XREAL_1:235; then A282: (len(F.k))-'1+(1+1) = (len(F.k))+1; A283: 1 <= (len(F.k))-'1 by A237,A279,NAT_D:49; then consider i1,j1,i2,j2 being Element of NAT such that A284: [i1,j1] in Indices G and A285: (F.k)/.((len(F.k)) -'1) = G*(i1,j1) and A286: [i2,j2] in Indices G and A287: (F.k)/.len(F.k) = G*(i2,j2) and i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A235,A281,JORDAN8:3; A288: i1 = i2 & j1+1 = j2 implies [i2+1,j2] in Indices G by A193,A235,A236 ,A279,A284,A285,A286,A287; A289: front_left_cell(F.k,(len(F.k))-'1,G) meets C & i1 = i2+1 & j1 = j2 implies [i2,j2-'1] in Indices G by A193,A235,A236,A279,A284,A285,A286,A287; A290: front_left_cell(F.k,(len(F.k))-'1,G) meets C & i1+1 = i2 & j1 = j2 implies [i2,j2+1] in Indices G by A193,A235,A236,A279,A284,A285,A286,A287; A291: front_left_cell(F.k,(len(F.k))-'1,G) meets C & i1 = i2 & j1+1 = j2 implies [i2-'1,j2] in Indices G by A193,A235,A236,A279,A284,A285,A286,A287; A292: front_right_cell(F.k,(len(F.k))-'1,G) meets C & i1 = i2 & j1 = j2+1 implies [i2,j2-'1] in Indices G by A193,A235,A236,A279,A284,A285,A286,A287 ; A293: i1 = i2+1 & j1 = j2 implies [i2,j2+1] in Indices G by A193,A235,A236 ,A279,A284,A285,A286,A287; A294: i1 +1 = i2 & j1 = j2 implies [i2,j2-'1] in Indices G by A193,A235,A236 ,A279,A284,A285,A286,A287; A295: 1 <= j2 by A286,MATRIX_1:38; A296: right_cell(F.k,(len(F.k))-'1,G) meets C by A236,A283,A281; A297: front_right_cell(F.k,(len(F.k))-'1,G) meets C & i1 = i2+1 & j1 = j2 implies [i2-'1,j2] in Indices G by A193,A235,A236,A279,A284,A285,A286,A287; A298: front_right_cell(F.k,(len(F.k))-'1,G) meets C & i1+1 = i2 & j1 = j2 implies [i2+1,j2] in Indices G by A193,A235,A236,A279,A284,A285,A286,A287; A299: front_right_cell(F.k,(len(F.k))-'1,G) meets C & i1 = i2 & j1+1 = j2 implies [i2,j2+1] in Indices G by A193,A235,A236,A279,A284,A285,A286,A287; A300: i1 = i2 & j1 = j2+1 implies [i2-'1,j2] in Indices G by A193,A235,A236 ,A279,A284,A285,A286,A287; A301: front_left_cell(F.k,(len(F.k))-'1,G) meets C & i1 = i2 & j1 = j2 +1 implies [i2+1,j2] in Indices G by A193,A235,A236,A279,A284,A285,A286,A287; (len(F.k))-'1 <= len(F.k) by NAT_D:35; then A302: (len(F.k))-'1 in dom(F.k) by A283,FINSEQ_3:25; A303: 1 <= i2 by A286,MATRIX_1:38; thus A304: F.(k+1) is_sequence_on G proof per cases; suppose front_left_cell(F.k,(len(F.k))-'1,G) misses C & front_right_cell(F.k,(len(F.k))-'1,G) misses C; then consider i,j such that A305: (F.k)^<*G*(i,j)*> turns_right (len(F.k))-'1,G and A306: F.(k+1) = (F.k)^<*G*(i,j)*> by A171,A235,A237,A279,A296; thus thesis proof set f = (F.k)^<*G*(i,j)*>; A307: f/.(len(F.k)+1) = G*(i,j) by FINSEQ_4:67; A308: f/.(len(F.k)-'1) = G*(i1,j1) & f/.len(F.k) = G*(i2,j2) by A285 ,A287,A302,A280,FINSEQ_4:68; per cases by A281,A284,A286,A282,A305,A308,GOBRD13:def 6; suppose that A309: i1 = i2 & j1+1 = j2 and A310: f/.(len(F.k)+1) = G*(i2+1,j2); now let i19,j19,i29,j29 be Element of NAT; assume A311: [i19,j19] in Indices G & [i29,j29] in Indices G & (F.k)/.len(F.k) = G* (i19,j19) & G*(i2+1,j2) = G*(i29,j29); then j2 = j19 & j2 = j29 by A286,A287,A288,A309,GOBOARD1:5; then A312: abs(j29-j19) = 0 by ABSVALUE:def 1; i2 = i19 & i2+1 = i29 by A286,A287,A288,A309,A311,GOBOARD1:5 ; hence abs(i29-i19)+abs(j29-j19) = 1 by A312,ABSVALUE:def 1; end; hence thesis by A235,A237,A279,A288,A306,A307,A309,A310, CARD_1:27,JORDAN8:6; end; suppose that A313: i1+1 = i2 & j1 = j2 and A314: f/.(len(F.k)+1) = G*(i2,j2-'1); now let i19,j19,i29,j29 be Element of NAT; assume A315: [i19,j19] in Indices G & [i29,j29] in Indices G & (F.k)/.len(F.k) = G* (i19,j19) & G*(i2,j2-'1) = G* (i29,j29); then j2 = j19 & j2-'1 = j29 by A286,A287,A294,A313,GOBOARD1:5 ; then j19-j29 = j2-(j2-1) by A295,XREAL_1:233; then A316: abs(j19-j29) = 1 by ABSVALUE:def 1; i2 = i19 & i2 = i29 by A286,A287,A294,A313,A315,GOBOARD1:5; then abs(i29-i19) = 0 by ABSVALUE:def 1; hence abs(i29-i19)+abs(j29-j19) = 1 by A316,UNIFORM1:11; end; hence thesis by A235,A237,A279,A294,A306,A307,A313,A314, CARD_1:27,JORDAN8:6; end; suppose that A317: i1 = i2+1 & j1 = j2 and A318: f/.(len(F.k)+1) = G*(i2,j2+1); now let i19,j19,i29,j29 be Element of NAT; assume A319: [i19,j19] in Indices G & [i29,j29] in Indices G & (F.k)/.len(F.k) = G* (i19,j19) & G*(i2,j2+1) = G*(i29,j29); then i2 = i19 & i2 = i29 by A286,A287,A293,A317,GOBOARD1:5; then A320: abs(i29-i19) = 0 by ABSVALUE:def 1; j2 = j19 & j2+1 = j29 by A286,A287,A293,A317,A319,GOBOARD1:5 ; hence abs(i29-i19)+abs(j29-j19) = 1 by A320,ABSVALUE:def 1; end; hence thesis by A235,A237,A279,A293,A306,A307,A317,A318, CARD_1:27,JORDAN8:6; end; suppose that A321: i1 = i2 & j1 = j2+1 and A322: f/.(len(F.k)+1) = G*(i2-'1,j2); now let i19,j19,i29,j29 be Element of NAT; assume A323: [i19,j19] in Indices G & [i29,j29] in Indices G & (F.k)/.len(F.k) = G* (i19,j19) & G*(i2-'1,j2) = G* (i29,j29); then i2 = i19 & i2-'1 = i29 by A286,A287,A300,A321,GOBOARD1:5 ; then i19-i29 = i2-(i2-1) by A303,XREAL_1:233; then A324: abs(i19-i29) = 1 by ABSVALUE:def 1; j2 = j19 & j2 = j29 by A286,A287,A300,A321,A323,GOBOARD1:5; then abs(j29-j19) = 0 by ABSVALUE:def 1; hence abs(i29-i19)+abs(j29-j19) = 1 by A324,UNIFORM1:11; end; hence thesis by A235,A237,A279,A300,A306,A307,A321,A322, CARD_1:27,JORDAN8:6; end; end; end; suppose A325: front_left_cell(F.k,(len(F.k))-'1,G) misses C & front_right_cell(F.k,(len(F.k))-'1,G) meets C; then consider i,j such that A326: (F.k)^<*G*(i,j)*> goes_straight (len(F.k))-'1,G and A327: F.(k+1) = (F.k)^<*G*(i,j)*> by A171,A235,A237,A279,A296; thus thesis proof set f = (F.k)^<*G*(i,j)*>; A328: f/.(len(F.k)+1) = G*(i,j) by FINSEQ_4:67; A329: f/.(len(F.k)-'1) = G*(i1,j1) & f/.len(F.k) = G*(i2,j2) by A285 ,A287,A302,A280,FINSEQ_4:68; per cases by A281,A284,A286,A282,A326,A329,GOBRD13:def 8; suppose that A330: i1 = i2 & j1+1 = j2 and A331: f/.(len(F.k)+1) = G*(i2,j2+1); now let i19,j19,i29,j29 be Element of NAT; assume A332: [i19,j19] in Indices G & [i29,j29] in Indices G & (F.k)/.len(F.k) = G* (i19,j19) & G*(i2,j2+1) = G*(i29,j29); then i2 = i19 & i2 = i29 by A286,A287,A299,A325,A330, GOBOARD1:5; then A333: abs(i29-i19) = 0 by ABSVALUE:def 1; j2 = j19 & j2+1 = j29 by A286,A287,A299,A325,A330,A332, GOBOARD1:5; hence abs(i29-i19)+abs(j29-j19) = 1 by A333,ABSVALUE:def 1; end; hence thesis by A235,A237,A279,A299,A325,A327,A328,A330,A331, CARD_1:27,JORDAN8:6; end; suppose that A334: i1+1 = i2 & j1 = j2 and A335: f/.(len(F.k)+1) = G*(i2+1,j2); now let i19,j19,i29,j29 be Element of NAT; assume A336: [i19,j19] in Indices G & [i29,j29] in Indices G & (F.k)/.len(F.k) = G* (i19,j19) & G*(i2+1,j2) = G* (i29,j29); then j2 = j19 & j2 = j29 by A286,A287,A298,A325,A334, GOBOARD1:5; then A337: abs(j29-j19) = 0 by ABSVALUE:def 1; i2 = i19 & i2+1 = i29 by A286,A287,A298,A325,A334,A336, GOBOARD1:5; hence abs(i29-i19)+abs(j29-j19) = 1 by A337,ABSVALUE:def 1; end; hence thesis by A235,A237,A279,A298,A325,A327,A328,A334,A335, CARD_1:27,JORDAN8:6; end; suppose that A338: i1 = i2+1 & j1 = j2 and A339: f/.(len(F.k)+1) = G*(i2-'1,j2); now let i19,j19,i29,j29 be Element of NAT; assume A340: [i19,j19] in Indices G & [i29,j29] in Indices G & (F.k)/.len(F.k) = G* (i19,j19) & G*(i2-'1,j2) = G*(i29,j29); then i2 = i19 & i2-'1 = i29 by A286,A287,A297,A325,A338, GOBOARD1:5; then i19-i29 = i2-(i2-1) by A303,XREAL_1:233; then A341: abs(i19-i29) = 1 by ABSVALUE:def 1; j2 = j19 & j2 = j29 by A286,A287,A297,A325,A338,A340, GOBOARD1:5; then abs(j29-j19) = 0 by ABSVALUE:def 1; hence abs(i29-i19)+abs(j29-j19) = 1 by A341,UNIFORM1:11; end; hence thesis by A235,A237,A279,A297,A325,A327,A328,A338,A339, CARD_1:27,JORDAN8:6; end; suppose that A342: i1 = i2 & j1 = j2+1 and A343: f/.(len(F.k)+1) = G*(i2,j2-'1); now let i19,j19,i29,j29 be Element of NAT; assume A344: [i19,j19] in Indices G & [i29,j29] in Indices G & (F.k)/.len(F.k) = G* (i19,j19) & G*(i2,j2-'1) = G* (i29,j29); then j2 = j19 & j2-'1 = j29 by A286,A287,A292,A325,A342, GOBOARD1:5; then j19-j29 = j2-(j2-1) by A295,XREAL_1:233; then A345: abs(j19-j29) = 1 by ABSVALUE:def 1; i2 = i19 & i2 = i29 by A286,A287,A292,A325,A342,A344, GOBOARD1:5; then abs(i29-i19) = 0 by ABSVALUE:def 1; hence abs(i29-i19)+abs(j29-j19) = 1 by A345,UNIFORM1:11; end; hence thesis by A235,A237,A279,A292,A325,A327,A328,A342,A343, CARD_1:27,JORDAN8:6; end; end; end; suppose A346: front_left_cell(F.k,(len(F.k))-'1,G) meets C; then consider i,j such that A347: (F.k)^<*G*(i,j)*> turns_left (len(F.k))-'1,G and A348: F.(k+1) = (F.k)^<*G*(i,j)*> by A171,A235,A237,A279,A296; thus thesis proof set f = (F.k)^<*G*(i,j)*>; A349: f/.(len(F.k)+1) = G*(i,j) by FINSEQ_4:67; A350: f/.(len(F.k)-'1) = G*(i1,j1) & f/.len(F.k) = G*(i2,j2) by A285 ,A287,A302,A280,FINSEQ_4:68; per cases by A281,A284,A286,A282,A347,A350,GOBRD13:def 7; suppose that A351: i1 = i2 & j1+1 = j2 and A352: f/.(len(F.k)+1) = G*(i2-'1,j2); now let i19,j19,i29,j29 be Element of NAT; assume A353: [i19,j19] in Indices G & [i29,j29] in Indices G & (F.k)/.len(F.k) = G* (i19,j19) & G*(i2-'1,j2) = G*(i29,j29); then i2 = i19 & i2-'1 = i29 by A286,A287,A291,A346,A351, GOBOARD1:5; then i19-i29 = i2-(i2-1) by A303,XREAL_1:233; then A354: abs(i19-i29) = 1 by ABSVALUE:def 1; j2 = j19 & j2 = j29 by A286,A287,A291,A346,A351,A353, GOBOARD1:5; then abs(j29-j19) = 0 by ABSVALUE:def 1; hence abs(i29-i19)+abs(j29-j19) = 1 by A354,UNIFORM1:11; end; hence thesis by A235,A237,A279,A291,A346,A348,A349,A351,A352, CARD_1:27,JORDAN8:6; end; suppose that A355: i1+1 = i2 & j1 = j2 and A356: f/.(len(F.k)+1) = G*(i2,j2+1); now let i19,j19,i29,j29 be Element of NAT; assume A357: [i19,j19] in Indices G & [i29,j29] in Indices G & (F.k)/.len(F.k) = G* (i19,j19) & G*(i2,j2+1) = G* (i29,j29); then i2 = i19 & i2 = i29 by A286,A287,A290,A346,A355, GOBOARD1:5; then A358: abs(i29-i19) = 0 by ABSVALUE:def 1; j2 = j19 & j2+1 = j29 by A286,A287,A290,A346,A355,A357, GOBOARD1:5; hence abs(i29-i19)+abs(j29-j19) = 1 by A358,ABSVALUE:def 1; end; hence thesis by A235,A237,A279,A290,A346,A348,A349,A355,A356, CARD_1:27,JORDAN8:6; end; suppose that A359: i1 = i2+1 & j1 = j2 and A360: f/.(len(F.k)+1) = G*(i2,j2-'1); now let i19,j19,i29,j29 be Element of NAT; assume A361: [i19,j19] in Indices G & [i29,j29] in Indices G & (F.k)/.len(F.k) = G* (i19,j19) & G*(i2,j2-'1) = G*(i29,j29); then j2 = j19 & j2-'1 = j29 by A286,A287,A289,A346,A359, GOBOARD1:5; then j19-j29 = j2-(j2-1) by A295,XREAL_1:233; then A362: abs(j19-j29) = 1 by ABSVALUE:def 1; i2 = i19 & i2 = i29 by A286,A287,A289,A346,A359,A361, GOBOARD1:5; then abs(i29-i19) = 0 by ABSVALUE:def 1; hence abs(i29-i19)+abs(j29-j19) = 1 by A362,UNIFORM1:11; end; hence thesis by A235,A237,A279,A289,A346,A348,A349,A359,A360, CARD_1:27,JORDAN8:6; end; suppose that A363: i1 = i2 & j1 = j2+1 and A364: f/.(len(F.k)+1) = G*(i2+1,j2); now let i19,j19,i29,j29 be Element of NAT; assume A365: [i19,j19] in Indices G & [i29,j29] in Indices G & (F.k)/.len(F.k) = G* (i19,j19) & G*(i2+1,j2) = G* (i29,j29); then j2 = j19 & j2 = j29 by A286,A287,A301,A346,A363, GOBOARD1:5; then A366: abs(j29-j19) = 0 by ABSVALUE:def 1; i2 = i19 & i2+1 = i29 by A286,A287,A301,A346,A363,A365, GOBOARD1:5; hence abs(i29-i19)+abs(j29-j19) = 1 by A366,ABSVALUE:def 1; end; hence thesis by A235,A237,A279,A301,A346,A348,A349,A363,A364, CARD_1:27,JORDAN8:6; end; end; end; end; let m such that A367: 1 <= m and A368: m+1 <= len(F.(k+1)); A369: left_cell(F.k,(len(F.k))-'1,G) misses C by A236,A283,A281; now per cases; suppose A370: m+1 = len(F.(k+1)); A371: (j2-'1)+1 = j2 by A295,XREAL_1:235; A372: (i2-'1)+1 = i2 by A303,XREAL_1:235; thus thesis proof per cases; suppose A373: front_left_cell(F.k,(len(F.k))-'1,G) misses C & front_right_cell(F.k,(len(F.k))-'1,G) misses C; then A374: ex i,j st (F.k)^<*G*(i,j)*> turns_right (len(F.k))-'1,G & F.(k+1) = (F.k)^<*G*(i,j)*> by A171,A235,A237,A279,A296; then A375: (F.(k+1))/.len(F.k) = G* (i2,j2) by A287,A280,FINSEQ_4:68; A376: (F.(k+1))/.(len(F.k)-'1) = G*(i1,j1) by A285,A302,A374, FINSEQ_4:68; now per cases by A281,A284,A286,A282,A374,A376,A375,GOBRD13:def 6 ; suppose that A377: i1 = i2 & j1+1 = j2 and A378: (F.(k+1))/.((len(F.k))+1) = G*(i2+1,j2); front_right_cell(F.k,(len(F.k))-'1,G) = cell(G,i1,j2 ) by A235,A283,A281,A284,A285,A286,A287,A377,GOBRD13:35; hence left_cell(F.(k+1),m,G) misses C by A237,A238,A286 ,A288,A304,A367,A370,A373,A375,A377,A378,GOBRD13:23; j2-'1 = j1 & cell(G,i1,j1) meets C by A235,A283,A281,A284 ,A285,A286,A287,A296,A377,GOBRD13:22,NAT_D:34; hence right_cell(F.(k+1),m,G) meets C by A237,A238,A286 ,A288,A304,A367,A370,A375,A377,A378,GOBRD13:24; end; suppose that A379: i1+1 = i2 & j1 = j2 and A380: (F.(k+1))/.((len(F.k))+1) = G*(i2,j2-'1); front_right_cell(F.k,(len(F.k))-'1,G) = cell(G,i2,j2 -'1) by A235,A283,A281,A284,A285,A286,A287,A379,GOBRD13:37; hence left_cell(F.(k+1),m,G) misses C by A237,A238,A286 ,A294,A304,A367,A370,A371,A373,A375,A379,A380,GOBRD13:27; i2-'1 = i1 & cell(G,i1,j1-'1) meets C by A235,A283,A281 ,A284,A285,A286,A287,A296,A379,GOBRD13:24,NAT_D:34; hence right_cell(F.(k+1),m,G) meets C by A237,A238,A286 ,A294,A304,A367,A370,A371,A375,A379,A380,GOBRD13:28; end; suppose that A381: i1 = i2+1 & j1 = j2 and A382: (F.(k+1))/.((len(F.k))+1) = G*(i2,j2+1); front_right_cell(F.k,(len(F.k))-'1,G) = cell(G,i2-'1 ,j2) by A235,A283,A281,A284,A285,A286,A287,A381,GOBRD13:39; hence left_cell(F.(k+1),m,G) misses C by A237,A238,A286 ,A293,A304,A367,A370,A373,A375,A381,A382,GOBRD13:21; cell(G,i2,j2) meets C by A235,A283,A281,A284,A285,A286,A287 ,A296,A381,GOBRD13:26; hence right_cell(F.(k+1),m,G) meets C by A237,A238,A286 ,A293,A304,A367,A370,A375,A381,A382,GOBRD13:22; end; suppose that A383: i1 = i2 & j1 = j2+1 and A384: (F.(k+1))/.((len(F.k))+1) = G*(i2-'1,j2); front_right_cell(F.k,(len(F.k))-'1,G) = cell(G,i2-'1 ,j2-'1 ) by A235,A283,A281,A284,A285,A286,A287,A383,GOBRD13:41; hence left_cell(F.(k+1),m,G) misses C by A237,A238,A286 ,A300,A304,A367,A370,A372,A373,A375,A383,A384,GOBRD13:25; cell(G,i2-'1,j2) meets C by A235,A283,A281,A284,A285,A286 ,A287,A296,A383,GOBRD13:28; hence right_cell(F.(k+1),m,G) meets C by A237,A238,A286 ,A300,A304,A367,A370,A372,A375,A383,A384,GOBRD13:26; end; end; hence thesis; end; suppose A385: front_left_cell(F.k,(len(F.k))-'1,G) misses C & front_right_cell(F.k,(len(F.k))-'1,G) meets C; then A386: ex i,j st (F.k)^<*G*(i,j)*> goes_straight (len(F.k))-'1,G & F.(k+1) = (F.k)^<*G*(i,j)*> by A171,A235,A237,A279,A296; then A387: (F.(k+1))/.len(F.k) = G*(i2,j2) by A287,A280,FINSEQ_4:68; A388: (F.(k+1))/.(len(F.k)-'1) = G*(i1,j1) by A285,A302,A386, FINSEQ_4:68; now per cases by A281,A284,A286,A282,A386,A388,A387,GOBRD13:def 8 ; suppose that A389: i1 = i2 & j1+1 = j2 and A390: (F.(k+1))/.(len(F.k)+1) = G*(i2,j2+1); front_left_cell(F.k,(len(F.k))-'1,G) = cell(G,i1-'1, j2) by A235,A283,A281,A284,A285,A286,A287,A389,GOBRD13:34; hence left_cell(F.(k+1),m,G) misses C by A237,A238,A286 ,A299,A304,A367,A370,A385,A387,A389,A390,GOBRD13:21; front_right_cell(F.k,(len(F.k))-'1,G) = cell(G,i1,j2 ) by A235,A283,A281,A284,A285,A286,A287,A389,GOBRD13:35; hence right_cell(F.(k+1),m,G) meets C by A237,A238,A286 ,A299,A304,A367,A370,A385,A387,A389,A390,GOBRD13:22; end; suppose that A391: i1+1 = i2 & j1 = j2 and A392: (F.(k+1))/.(len(F.k)+1) = G*(i2+1,j2); front_left_cell(F.k,(len(F.k))-'1,G) = cell(G,i2,j2) by A235,A283,A281,A284,A285,A286,A287,A391,GOBRD13:36; hence left_cell(F.(k+1),m,G) misses C by A237,A238,A286 ,A298,A304,A367,A370,A385,A387,A391,A392,GOBRD13:23; front_right_cell(F.k,(len(F.k))-'1,G) = cell(G,i2,j2 -'1) by A235,A283,A281,A284,A285,A286,A287,A391,GOBRD13:37; hence right_cell(F.(k+1),m,G) meets C by A237,A238,A286 ,A298,A304,A367,A370,A385,A387,A391,A392,GOBRD13:24; end; suppose that A393: i1 = i2+1 & j1 = j2 and A394: (F.(k+1))/.(len(F.k)+1) = G*(i2-'1,j2); front_left_cell(F.k,(len(F.k))-'1,G) = cell(G,i2-'1, j2-'1) by A235,A283,A281,A284,A285,A286,A287,A393,GOBRD13:38; hence left_cell(F.(k+1),m,G) misses C by A237,A238,A286 ,A297,A304,A367,A370,A372,A385,A387,A393,A394,GOBRD13:25; front_right_cell(F.k,(len(F.k))-'1,G) = cell(G,i2-'1 ,j2) by A235,A283,A281,A284,A285,A286,A287,A393,GOBRD13:39; hence right_cell(F.(k+1),m,G) meets C by A237,A238,A286 ,A297,A304,A367,A370,A372,A385,A387,A393,A394,GOBRD13:26; end; suppose that A395: i1 = i2 & j1 = j2+1 and A396: (F.(k+1))/.(len(F.k)+1) = G*(i2,j2-'1); front_left_cell(F.k,(len(F.k))-'1,G) = cell(G,i2,j2 -'1) by A235,A283,A281,A284,A285,A286,A287,A395,GOBRD13:40; hence left_cell(F.(k+1),m,G) misses C by A237,A238,A286 ,A292,A304,A367,A370,A371,A385,A387,A395,A396,GOBRD13:27; front_right_cell(F.k,(len(F.k))-'1,G) = cell(G,i2-'1 ,j2-'1 ) by A235,A283,A281,A284,A285,A286,A287,A395,GOBRD13:41; hence right_cell(F.(k+1),m,G) meets C by A237,A238,A286 ,A292,A304,A367,A370,A371,A385,A387,A395,A396,GOBRD13:28; end; end; hence thesis; end; suppose A397: front_left_cell(F.k,(len(F.k))-'1,G) meets C; then A398: ex i,j st (F.k)^<*G*(i,j)*> turns_left (len(F.k))-'1,G & F.(k+1) = (F.k)^<*G*(i,j)*> by A171,A235,A237,A279,A296; then A399: (F.(k+1))/.len(F.k) = G* (i2,j2) by A287,A280,FINSEQ_4:68; A400: (F.(k+1))/.(len(F.k)-'1) = G*(i1,j1) by A285,A302,A398, FINSEQ_4:68; now per cases by A281,A284,A286,A282,A398,A400,A399,GOBRD13:def 7 ; suppose that A401: i1 = i2 & j1+1 = j2 and A402: (F.(k+1))/.(len(F.k)+1) = G*(i2-'1,j2); j2-'1 = j1 & cell(G,i1-'1,j1) misses C by A235,A283,A281 ,A284,A285,A286,A287,A369,A401,GOBRD13:21,NAT_D:34; hence left_cell(F.(k+1),m,G) misses C by A237,A238,A286 ,A291,A304,A367,A370,A372,A397,A399,A401,A402,GOBRD13:25; front_left_cell(F.k,(len(F.k))-'1,G) = cell(G,i1-'1, j2) by A235,A283,A281,A284,A285,A286,A287,A401,GOBRD13:34; hence right_cell(F.(k+1),m,G) meets C by A237,A238,A286 ,A291,A304,A367,A370,A372,A397,A399,A401,A402,GOBRD13:26; end; suppose that A403: i1+1 = i2 & j1 = j2 and A404: (F.(k+1))/.(len(F.k)+1) = G*(i2,j2+1); i2-'1 = i1 & cell(G,i1,j1) misses C by A235,A283,A281,A284 ,A285,A286,A287,A369,A403,GOBRD13:23,NAT_D:34; hence left_cell(F.(k+1),m,G) misses C by A237,A238,A286 ,A290,A304,A367,A370,A397,A399,A403,A404,GOBRD13:21; front_left_cell(F.k,(len(F.k))-'1,G) = cell(G,i2,j2) by A235,A283,A281,A284,A285,A286,A287,A403,GOBRD13:36; hence right_cell(F.(k+1),m,G) meets C by A237,A238,A286 ,A290,A304,A367,A370,A397,A399,A403,A404,GOBRD13:22; end; suppose that A405: i1 = i2+1 & j1 = j2 and A406: (F.(k+1))/.(len(F.k)+1) = G*(i2,j2-'1); cell(G,i2,j2-'1) misses C by A235,A283,A281,A284,A285,A286 ,A287,A369,A405,GOBRD13:25; hence left_cell(F.(k+1),m,G) misses C by A237,A238,A286 ,A289,A304,A367,A370,A371,A397,A399,A405,A406,GOBRD13:27; front_left_cell(F.k,(len(F.k))-'1,G) = cell(G,i2-'1, j2-'1) by A235,A283,A281,A284,A285,A286,A287,A405,GOBRD13:38; hence right_cell(F.(k+1),m,G) meets C by A237,A238,A286 ,A289,A304,A367,A370,A371,A397,A399,A405,A406,GOBRD13:28; end; suppose that A407: i1 = i2 & j1 = j2+1 and A408: (F.(k+1))/.(len(F.k)+1) = G*(i2+1,j2); cell(G,i2,j2) misses C by A235,A283,A281,A284,A285,A286 ,A287,A369,A407,GOBRD13:27; hence left_cell(F.(k+1),m,G) misses C by A237,A238,A286 ,A301,A304,A367,A370,A397,A399,A407,A408,GOBRD13:23; front_left_cell(F.k,(len(F.k))-'1,G) = cell(G,i2,j2 -'1) by A235,A283,A281,A284,A285,A286,A287,A407,GOBRD13:40; hence right_cell(F.(k+1),m,G) meets C by A237,A238,A286 ,A301,A304,A367,A370,A397,A399,A407,A408,GOBRD13:24; end; end; hence thesis; end; end; end; suppose m+1 <> len(F.(k+1)); then m+1 < len(F.(k+1)) by A368,XXREAL_0:1; then A409: m+1 <= len(F.k)by A237,A238,NAT_1:13; then consider i1,j1,i2,j2 being Element of NAT such that A410: [i1,j1] in Indices G and A411: (F.k)/.m = G*(i1,j1) and A412: [i2,j2] in Indices G and A413: (F.k)/.(m+1) = G*(i2,j2) and A414: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A235,A367,JORDAN8:3; A415: left_cell(F.k,m,G) misses C & right_cell(F.k,m,G) meets C by A236 ,A367,A409; A416: now per cases; suppose front_left_cell(F.k,(len(F.k))-'1,G) misses C & front_right_cell(F.k,(len(F.k))-'1,G) misses C; then consider i,j such that (F.k)^<*G*(i,j)*> turns_right (len(F.k))-'1,G and A417: F.(k+1) = (F.k)^<*G*(i,j)*> by A171,A235,A237,A279,A296; take i,j; thus F.(k+1) = (F.k)^<*G*(i,j)*> by A417; end; suppose front_left_cell(F.k,(len(F.k))-'1,G) misses C & front_right_cell(F.k,(len(F.k))-'1,G) meets C; then consider i,j such that (F.k)^<*G*(i,j)*> goes_straight (len(F.k))-'1,G and A418: F.(k+1) = (F.k)^<*G*(i,j)*> by A171,A235,A237,A279,A296; take i,j; thus F.(k+1) = (F.k)^<*G*(i,j)*> by A418; end; suppose front_left_cell(F.k,(len(F.k))-'1,G) meets C; then consider i,j such that (F.k)^<*G*(i,j)*> turns_left (len(F.k))-'1,G and A419: F.(k+1) = (F.k)^<*G*(i,j)*> by A171,A235,A237,A279,A296; take i,j; thus F.(k+1) = (F.k)^<*G*(i,j)*> by A419; end; end; 1 <= m+1 by NAT_1:12; then m+1 in dom(F.k) by A409,FINSEQ_3:25; then A420: (F.(k+1))/.(m+1) = G*(i2,j2) by A413,A416,FINSEQ_4:68; m <= len(F.k) by A409,NAT_1:13; then m in dom(F.k) by A367,FINSEQ_3:25; then A421: (F.(k+1))/.m = G*(i1,j1) by A411,A416,FINSEQ_4:68; now per cases by A414; suppose A422: i1 = i2 & j1+1 = j2; then left_cell(F.k,m,G) = cell(G,i1-'1,j1) & right_cell(F.k,m ,G) = cell(G,i1,j1) by A235,A367,A409,A410,A411,A412,A413,GOBRD13:21,22; hence thesis by A304,A367,A368,A410,A412,A415,A421,A420,A422, GOBRD13:21,22; end; suppose A423: i1+1 = i2 & j1 = j2; then left_cell(F.k,m,G) = cell(G,i1,j1) & right_cell(F.k,m,G) = cell(G,i1,j1-'1) by A235,A367,A409,A410,A411,A412,A413,GOBRD13:23,24; hence thesis by A304,A367,A368,A410,A412,A415,A421,A420,A423, GOBRD13:23,24; end; suppose A424: i1 = i2+1 & j1 = j2; then left_cell(F.k,m,G) = cell(G,i2,j2-'1) & right_cell(F.k,m ,G) = cell(G,i2,j2) by A235,A367,A409,A410,A411,A412,A413,GOBRD13:25,26; hence thesis by A304,A367,A368,A410,A412,A415,A421,A420,A424, GOBRD13:25,26; end; suppose A425: i1 = i2 & j1 = j2+1; then left_cell(F.k,m,G) = cell(G,i2,j2) & right_cell(F.k,m,G) = cell(G,i1-'1,j2) by A235,A367,A409,A410,A411,A412,A413,GOBRD13:27,28; hence thesis by A304,A367,A368,A410,A412,A415,A421,A420,A425, GOBRD13:27,28; end; end; hence thesis; end; end; hence thesis; end; end; defpred Q[Nat] means ex w being Element of NAT st w = $1 & $1 >= 1 & ex m st m in dom(F.w) & m <> len(F.w) & (F.w)/.m = (F.w)/.len(F.w); A426: P[0] proof ( for n st n in dom(F.0) ex i,j st [i,j] in Indices G & (F.0)/.n = G*(i,j))& for n st n in dom(F.0) & n+1 in dom(F.0) holds for m,k,i,j st [m,k] in Indices G & [i,j] in Indices G & (F.0)/.n = G*(m,k) & (F.0)/.(n+1) = G*(i,j) holds abs( m-i)+abs(k-j) = 1 by A170; hence F.0 is_sequence_on G by GOBOARD1:def 9; let m; assume that 1 <= m and A427: m+1 <= len(F.0); thus thesis by A170,A427,CARD_1:27; end; A428: for k holds P[k] from NAT_1:sch 1(A426,A232); A429: for k,i1,i2,j1,j2 st k > 1 & [i1,j1] in Indices G & (F.k)/.((len(F.k )) -'1) = G*(i1,j1) & [i2,j2] in Indices G & (F.k)/.len(F.k) = G*(i2,j2) holds (front_left_cell(F.k,(len(F.k))-'1,G) misses C & front_right_cell(F.k,(len(F.k) )-'1,G) misses C implies F.(k+1) turns_right (len(F.k))-'1,G & (i1 = i2 & j1+1 = j2 implies [i2+1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2+1,j2)*>) & (i1+1 = i2 & j1 = j2 implies [i2,j2-'1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2-'1) *>) & (i1 = i2+1 & j1 = j2 implies [i2,j2+1] in Indices G & F.(k+1) = (F.k)^<*G *(i2,j2+1)*>)& (i1 = i2 & j1 = j2+1 implies [i2-'1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2-'1,j2)*>)) & (front_left_cell(F.k,(len(F.k))-'1,G) misses C & front_right_cell(F.k,(len(F.k))-'1,G) meets C implies F.(k+1) goes_straight ( len(F.k))-'1,G & (i1 = i2 & j1+1 = j2 implies [i2,j2+1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2+1)*>) & (i1+1 = i2 & j1 = j2 implies [i2+1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2+1,j2)*>) & (i1 = i2+1 & j1 = j2 implies [i2-'1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2-'1,j2)*>)& (i1 = i2 & j1 = j2+1 implies [i2 ,j2-'1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2-'1)*>)) & (front_left_cell(F. k,(len(F.k))-'1,G) meets C implies F.(k+1) turns_left (len(F.k))-'1,G & (i1 = i2 & j1+1 = j2 implies [i2-'1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2-'1,j2) *>) & (i1+1 = i2 & j1 = j2 implies [i2,j2+1] in Indices G & F.(k+1) = (F.k)^<*G *(i2,j2+1)*>) & (i1 = i2+1 & j1 = j2 implies [i2,j2-'1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2-'1)*>)& (i1 = i2 & j1 = j2+1 implies [i2+1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2+1,j2)*>)) proof let k,i1,i2,j1,j2 such that A430: k > 1 and A431: [i1,j1] in Indices G and A432: (F.k)/.((len(F.k)) -'1) = G*(i1,j1) and A433: [i2,j2] in Indices G and A434: (F.k)/.len(F.k) = G*(i2,j2); A435: len(F.k) = k by A192; then A436: (len(F.k)) -'1 +1 = len(F.k) by A430,XREAL_1:235; then A437: (len(F.k))-'1+(1+1) = (len(F.k))+1; A438: 1 <= (len(F.k))-'1 by A430,A435,NAT_D:49; then A439: right_cell(F.k,(len(F.k))-'1,G) meets C by A428,A436; (len(F.k))-'1 <= len(F.k) by NAT_D:35; then A440: (len(F.k))-'1 in dom(F.k) by A438,FINSEQ_3:25; A441: j1+1 > j1 & j2+1 > j2 by NAT_1:13; A442: F.k is_sequence_on G by A428; A443: i1+1 > i1 & i2+1 > i2 by NAT_1:13; A444: len(F.k) in dom(F.k) by A430,A435,FINSEQ_3:25; hereby assume front_left_cell(F.k,(len(F.k))-'1,G) misses C & front_right_cell(F.k,(len(F. k))-'1,G) misses C; then consider i,j such that A445: (F.k)^<*G*(i,j)*> turns_right (len(F.k))-'1,G and A446: F.(k+1) = (F.k)^<*G*(i,j)*> by A171,A430,A442,A435,A439; thus F.(k+1) turns_right (len(F.k))-'1,G by A445,A446; A447: (F.(k+1))/.(len(F.k)+1) = G*(i,j) by A446,FINSEQ_4:67; A448: (F.(k+1))/.(len(F.k)-'1) = G*(i1,j1) & (F.(k+1))/.len(F.k) = G* (i2,j2) by A432,A434,A440,A444,A446,FINSEQ_4:68; hence i1 = i2 & j1+1 = j2 implies [i2+1,j2] in Indices G & F.(k+1) = (F .k)^<*G*(i2+1,j2)*> by A431,A433,A436,A437,A441,A445,A446,A447,GOBRD13:def 6; thus i1+1 = i2 & j1 = j2 implies [i2,j2-'1] in Indices G & F.(k+1) = ( F.k)^<*G*(i2,j2-'1)*> by A431,A433,A436,A437,A443,A445,A446,A448,A447, GOBRD13:def 6; thus i1 = i2+1 & j1 = j2 implies [i2,j2+1] in Indices G & F.(k+1) = (F .k)^<*G*(i2,j2+1)*> by A431,A433,A436,A437,A443,A445,A446,A448,A447, GOBRD13:def 6; thus i1 = i2 & j1 = j2+1 implies [i2-'1,j2] in Indices G & F.(k+1) = ( F.k)^<*G*(i2-'1,j2)*> by A431,A433,A436,A437,A441,A445,A446,A448,A447, GOBRD13:def 6; end; hereby assume front_left_cell(F.k,(len(F.k))-'1,G) misses C & front_right_cell(F.k,(len(F. k))-'1,G) meets C; then consider i,j such that A449: (F.k)^<*G*(i,j)*> goes_straight (len(F.k))-'1,G and A450: F.(k+1) = (F.k)^<*G*(i,j)*> by A171,A430,A442,A435,A439; thus F.(k+1) goes_straight (len(F.k))-'1,G by A449,A450; A451: (F.(k+1))/.(len(F.k)+1) = G*(i,j) by A450,FINSEQ_4:67; A452: (F.(k+1))/.(len(F.k)-'1) = G*(i1,j1) & (F.(k+1))/.len(F.k) = G* (i2,j2) by A432,A434,A440,A444,A450,FINSEQ_4:68; hence i1 = i2 & j1+1 = j2 implies [i2,j2+1] in Indices G & F.(k+1) = (F .k)^<*G*(i2,j2+1)*> by A431,A433,A436,A437,A441,A449,A450,A451,GOBRD13:def 8; thus i1+1 = i2 & j1 = j2 implies [i2+1,j2] in Indices G & F.(k+1) = (F .k)^<*G*(i2+1,j2)*> by A431,A433,A436,A437,A443,A449,A450,A452,A451, GOBRD13:def 8; thus i1 = i2+1 & j1 = j2 implies [i2-'1,j2] in Indices G & F.(k+1) = ( F.k)^<*G*(i2-'1,j2)*> by A431,A433,A436,A437,A443,A449,A450,A452,A451, GOBRD13:def 8; thus i1 = i2 & j1 = j2+1 implies [i2,j2-'1] in Indices G & F.(k+1) = ( F.k)^<*G*(i2,j2-'1)*> by A431,A433,A436,A437,A441,A449,A450,A452,A451, GOBRD13:def 8; end; assume front_left_cell(F.k,(len(F.k))-'1,G) meets C; then consider i,j such that A453: (F.k)^<*G*(i,j)*> turns_left (len(F.k))-'1,G and A454: F.(k+1) = (F.k)^<*G*(i,j)*> by A171,A430,A442,A435,A439; A455: (F.(k+1))/.len(F.k) = G* (i2,j2) by A434,A444,A454,FINSEQ_4:68; thus F.(k+1) turns_left (len(F.k))-'1,G by A453,A454; A456: (F.(k+1))/.(len(F.k)+1) = G*(i,j) by A454,FINSEQ_4:67; A457: (F.(k+1))/.(len(F.k)-'1) = G*(i1,j1) by A432,A440,A454,FINSEQ_4:68; hence i1 = i2 & j1+1 = j2 implies [i2-'1,j2] in Indices G & F.(k+1) = (F. k)^<*G*(i2-'1,j2)*> by A431,A433,A436,A437,A441,A453,A454,A455,A456, GOBRD13:def 7; thus i1+1 = i2 & j1 = j2 implies [i2,j2+1] in Indices G & F.(k+1) = (F.k )^<*G*(i2,j2+1)*> by A431,A433,A436,A437,A443,A453,A454,A457,A455,A456, GOBRD13:def 7; thus i1 = i2+1 & j1 = j2 implies [i2,j2-'1] in Indices G & F.(k+1) = (F. k)^<*G*(i2,j2-'1)*> by A431,A433,A436,A437,A443,A453,A454,A457,A455,A456, GOBRD13:def 7; thus thesis by A431,A433,A436,A437,A441,A453,A454,A457,A455,A456, GOBRD13:def 7; end; A458: for k st k > 1 holds (front_left_cell(F.k,k-'1,Gauge(C,n)) misses C & front_right_cell(F.k,k-'1,Gauge(C,n)) misses C implies F.(k+1) turns_right k -'1,Gauge(C,n)) & (front_left_cell(F.k,k-'1,Gauge(C,n)) misses C & front_right_cell(F.k,k-'1,Gauge(C,n)) meets C implies F.(k+1) goes_straight k-' 1,Gauge(C,n)) & (front_left_cell(F.k,k-'1,Gauge(C,n)) meets C implies F.(k+1) turns_left k-'1,Gauge(C,n)) proof let k such that A459: k > 1; A460: F.k is_sequence_on G by A428; A461: len(F.k) = k by A192; then 1 <= (len(F.k))-'1 & (len(F.k)) -'1 +1 = len(F.k) by A459,NAT_D:49 ,XREAL_1:235; then ex i1,j1,i2,j2 being Element of NAT st [i1,j1] in Indices G & (F.k )/.((len(F.k)) -'1) = G*(i1,j1) & [i2,j2] in Indices G & (F.k)/.len(F.k) = G*( i2,j2) & (i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1) by A460,JORDAN8:3; hence thesis by A429,A459,A461; end; defpred P[Element of NAT] means for m st m <= $1 holds (F.$1)|m = F.m; A462: P[0] proof let m; assume A463: m <= 0; then (F.0)|0 = (F.0)|m; hence thesis by A170,A463; end; A464: for k holds ex i,j st [i,j] in Indices G & F.(k+1) = (F.k)^<*G*(i,j) *> proof let k; A465: F.k is_sequence_on G by A428; A466: len(F.k) = k by A192; len G >= 4 by JORDAN8:10; then A467: len G = width G & 1 < len G by JORDAN8:def 1,XXREAL_0:2; per cases by XXREAL_0:1; suppose A468: k < 1; consider i such that A469: 1 <= i and A470: i+1 <= len G and N-min C in cell(G,i,width G-'1) and N-min C <> G*(i,width G-'1) and A471: F.(0+1) = <*G*(i,width G)*> by A171; take i, j = width G; i < len G by A470,NAT_1:13; hence [i,j] in Indices G by A467,A469,MATRIX_1:36; k = 0 by A468,NAT_1:14; hence thesis by A170,A471,FINSEQ_1:34; end; suppose A472: k = 1; consider i such that A473: 1 <= i and A474: i+1 <= len G and A475: N-min C in cell(G,i,width G-'1) & N-min C <> G*(i,width G-'1 ) and A476: F.(0+1) = <*G*(i,width G)*> by A171; take i+1,j = width G; 1 <= i+1 by A473,NAT_1:13; hence [i+1,j] in Indices G by A467,A474,MATRIX_1:36; consider i9 being Element of NAT such that A477: 1 <= i9 & i9+1 <= len G & N-min C in cell(G,i9,width G-'1) & N-min C <> G*(i9,width G-'1) and A478: F.(1+1) = <*G*(i9,width G),G*(i9+1,width G)*> by A171; i = i9 by A473,A474,A475,A477,Th29; hence thesis by A472,A476,A478,FINSEQ_1:def 9; end; suppose A479: k > 1; then 1 <= (len(F.k))-'1 & (len(F.k)) -'1 +1 = len(F.k) by A466,NAT_D:49 ,XREAL_1:235; then consider i1,j1,i2,j2 being Element of NAT such that A480: [i1,j1] in Indices G & (F.k)/.((len(F.k)) -'1) = G*(i1,j1) & [i2,j2] in Indices G & (F.k)/.len(F.k) = G*(i2,j2) and A481: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A465,JORDAN8:3; now per cases; suppose A482: front_left_cell(F.k,(len(F.k))-'1,G) misses C & front_right_cell(F.k,(len(F.k))-'1,G) misses C; now per cases by A481; suppose i1 = i2 & j1+1 = j2; then [i2+1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2+1,j2)*> by A429,A479,A480,A482; hence thesis; end; suppose i1+1 = i2 & j1 = j2; then [i2,j2-'1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2-'1) *> by A429,A479,A480,A482; hence thesis; end; suppose i1 = i2+1 & j1 = j2; then [i2,j2+1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2+1)*> by A429,A479,A480,A482; hence thesis; end; suppose i1 = i2 & j1 = j2+1; then [i2-'1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2-'1,j2) *> by A429,A479,A480,A482; hence thesis; end; end; hence thesis; end; suppose A483: front_left_cell(F.k,(len(F.k))-'1,G) misses C & front_right_cell(F.k,(len(F.k))-'1,G) meets C; now per cases by A481; suppose i1 = i2 & j1+1 = j2; then [i2,j2+1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2+1)*> by A429,A479,A480,A483; hence thesis; end; suppose i1+1 = i2 & j1 = j2; then [i2+1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2+1,j2)*> by A429,A479,A480,A483; hence thesis; end; suppose i1 = i2+1 & j1 = j2; then [i2-'1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2-'1,j2) *> by A429,A479,A480,A483; hence thesis; end; suppose i1 = i2 & j1 = j2+1; then [i2,j2-'1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2-'1) *> by A429,A479,A480,A483; hence thesis; end; end; hence thesis; end; suppose A484: front_left_cell(F.k,(len(F.k))-'1,G) meets C; now per cases by A481; suppose i1 = i2 & j1+1 = j2; then [i2-'1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2-'1,j2) *> by A429,A479,A480,A484; hence thesis; end; suppose i1+1 = i2 & j1 = j2; then [i2,j2+1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2+1)*> by A429,A479,A480,A484; hence thesis; end; suppose i1 = i2+1 & j1 = j2; then [i2,j2-'1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2-'1) *> by A429,A479,A480,A484; hence thesis; end; suppose i1 = i2 & j1 = j2+1; then [i2+1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2+1,j2)*> by A429,A479,A480,A484; hence thesis; end; end; hence thesis; end; end; hence thesis; end; end; A485: for k st P[k] holds P[k+1] proof let k such that A486: for m st m <= k holds (F.k)|m = F.m; let m such that A487: m <= k+1; per cases by A487,XXREAL_0:1; suppose m < k+1; then A488: m <= k by NAT_1:13; len(F.k) = k & ex i,j st [i,j] in Indices G & F.(k+1) = F.k^<*G* (i,j)*> by A192,A464; then (F.(k+1))|m = (F.k)|m by A488,FINSEQ_5:22; hence thesis by A486,A488; end; suppose A489: m = k+1; len(F.(k+1)) = k+1 by A192; hence thesis by A489,FINSEQ_1:58; end; end; A490: for k holds P[k] from NAT_1:sch 1(A462,A485); defpred P[Element of NAT] means F.$1 is unfolded; A491: for k st P[k] holds P[k+1] proof let k such that A492: F.k is unfolded; A493: F.k is_sequence_on G by A428; per cases; suppose k <= 1; then k+1 <= 1+1 by XREAL_1:6; then len(F.(k+1)) <= 2 by A192; hence thesis by SPPOL_2:26; end; suppose A494: k > 1; set m = k-'1; A495: m+1 = k by A494,XREAL_1:235; A496: len(F.k) = k by A192; A497: 1 <= m by A494,NAT_D:49; then consider i1,j1,i2,j2 being Element of NAT such that A498: [i1,j1] in Indices G and A499: (F.k)/.m = G*(i1,j1) and A500: [i2,j2] in Indices G and A501: (F.k)/.len(F.k) = G*(i2,j2) and A502: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A493,A495,A496,JORDAN8:3; A503: LSeg(F.k,m) = LSeg(G*(i1,j1),G*(i2,j2)) by A497,A495,A496,A499,A501, TOPREAL1:def 3; A504: 1 <= j2 by A500,MATRIX_1:38; then A505: (j2-'1)+1 = j2 by XREAL_1:235; A506: 1 <= j1 by A498,MATRIX_1:38; A507: 1 <= i2 by A500,MATRIX_1:38; then A508: (i2-'1)+1 = i2 by XREAL_1:235; A509: i1 <= len G by A498,MATRIX_1:38; A510: j2 <= width G by A500,MATRIX_1:38; A511: 1 <= i1 by A498,MATRIX_1:38; A512: j1 <= width G by A498,MATRIX_1:38; A513: i2 <= len G by A500,MATRIX_1:38; now per cases; suppose A514: front_left_cell(F.k,(len(F.k))-'1,G) misses C & front_right_cell(F.k,(len(F.k))-'1,G) misses C; now per cases by A502; suppose A515: i1 = i2 & j1+1 = j2; then [i2+1,j2] in Indices G by A429,A494,A496,A498,A499,A500 ,A501,A514; then i2+1 <= len G by MATRIX_1:38; then A516: LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G* (i2+1,j2)) = {(F. k)/.len(F.k)} by A501,A511,A506,A510,A503,A515,GOBOARD7:15; F.(k+1) = (F.k)^ <*G*(i2+1,j2)*> by A429,A494,A496,A498,A499 ,A500,A501,A514,A515; hence thesis by A492,A495,A496,A516,SPPOL_2:30; end; suppose A517: i1+1 = i2 & j1 = j2; then [i2,j2-'1] in Indices G by A429,A494,A496,A498,A499,A500 ,A501,A514; then 1 <= j2-'1 by MATRIX_1:38; then A518: LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G*(i2,j2-'1)) = {(F. k)/.len(F.k)} by A501,A511,A512,A513,A505,A503,A517,GOBOARD7:16; F.(k+1) = (F.k)^ <*G*(i2,j2-'1)*> by A429,A494,A496,A498,A499 ,A500,A501,A514,A517; hence thesis by A492,A495,A496,A518,SPPOL_2:30; end; suppose A519: i1 = i2+1 & j1 = j2; then [i2,j2+1] in Indices G by A429,A494,A496,A498,A499,A500 ,A501,A514; then j2+1 <= width G by MATRIX_1:38; then A520: LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G* (i2,j2+1)) = {(F. k)/.len(F.k)} by A501,A509,A506,A507,A503,A519,GOBOARD7:17; F.(k+1) = (F.k)^ <*G*(i2,j2+1)*> by A429,A494,A496,A498,A499 ,A500,A501,A514,A519; hence thesis by A492,A495,A496,A520,SPPOL_2:30; end; suppose A521: i1 = i2 & j1 = j2+1; then [i2-'1,j2] in Indices G by A429,A494,A496,A498,A499,A500 ,A501,A514; then 1 <= i2-'1 by MATRIX_1:38; then A522: LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G*(i2-'1,j2)) = {(F. k)/.len(F.k)} by A501,A509,A512,A504,A508,A503,A521,GOBOARD7:18; F.(k+1) = (F.k)^ <*G*(i2-'1,j2)*> by A429,A494,A496,A498,A499 ,A500,A501,A514,A521; hence thesis by A492,A495,A496,A522,SPPOL_2:30; end; end; hence thesis; end; suppose A523: front_left_cell(F.k,(len(F.k))-'1,G) misses C & front_right_cell(F.k,(len(F.k))-'1,G) meets C; now per cases by A502; suppose A524: i1 = i2 & j1+1 = j2; then [i2,j2+1] in Indices G by A429,A494,A496,A498,A499,A500 ,A501,A523; then A525: j2+1 <= width G by MATRIX_1:38; j2+1 = j1+(1+1) by A524; then A526: LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G* (i2,j2+1)) = {(F. k)/.len(F.k)} by A501,A511,A509,A506,A503,A524,A525,GOBOARD7:13; F.(k+1) = (F.k)^ <*G*(i2,j2+1)*> by A429,A494,A496,A498,A499 ,A500,A501,A523,A524; hence thesis by A492,A495,A496,A526,SPPOL_2:30; end; suppose A527: i1+1 = i2 & j1 = j2; then [i2+1,j2] in Indices G by A429,A494,A496,A498,A499,A500 ,A501,A523; then A528: i2+1 <= len G by MATRIX_1:38; i2+1 = i1+(1+1) by A527; then A529: LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G* (i2+1,j2)) = {(F. k)/.len(F.k)} by A501,A511,A506,A512,A503,A527,A528,GOBOARD7:14; F.(k+1) = (F.k)^ <*G*(i2+1,j2)*> by A429,A494,A496,A498,A499 ,A500,A501,A523,A527; hence thesis by A492,A495,A496,A529,SPPOL_2:30; end; suppose A530: i1 = i2+1 & j1 = j2; then [i2-'1,j2] in Indices G by A429,A494,A496,A498,A499,A500 ,A501,A523; then A531: 1 <= i2-'1 by MATRIX_1:38; i2-'1+1+1 = i2-'1+(1+1); then A532: LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G*(i2-'1,j2)) = {(F. k)/.len(F.k)} by A501,A509,A506,A512,A508,A503,A530,A531,GOBOARD7:14; F.(k+1) = (F.k)^ <*G*(i2-'1,j2)*> by A429,A494,A496,A498,A499 ,A500,A501,A523,A530; hence thesis by A492,A495,A496,A532,SPPOL_2:30; end; suppose A533: i1 = i2 & j1 = j2+1; then [i2,j2-'1] in Indices G by A429,A494,A496,A498,A499,A500 ,A501,A523; then A534: 1 <= j2-'1 by MATRIX_1:38; j2-'1+1+1 = j2-'1+(1+1); then A535: LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G*(i2,j2-'1)) = {(F. k)/.len(F.k)} by A501,A511,A509,A512,A505,A503,A533,A534,GOBOARD7:13; F.(k+1) = (F.k)^ <*G*(i2,j2-'1)*> by A429,A494,A496,A498,A499 ,A500,A501,A523,A533; hence thesis by A492,A495,A496,A535,SPPOL_2:30; end; end; hence thesis; end; suppose A536: front_left_cell(F.k,(len(F.k))-'1,G) meets C; now per cases by A502; suppose A537: i1 = i2 & j1+1 = j2; then [i2-'1,j2] in Indices G by A429,A494,A496,A498,A499,A500 ,A501,A536; then 1 <= i2-'1 by MATRIX_1:38; then A538: LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G*(i2-'1,j2)) = {(F. k)/.len(F.k)} by A501,A509,A506,A510,A508,A503,A537,GOBOARD7:16; F.(k+1) = (F.k)^ <*G*(i2-'1,j2)*> by A429,A494,A496,A498,A499 ,A500,A501,A536,A537; hence thesis by A492,A495,A496,A538,SPPOL_2:30; end; suppose A539: i1+1 = i2 & j1 = j2; then [i2,j2+1] in Indices G by A429,A494,A496,A498,A499,A500 ,A501,A536; then j2+1 <= width G by MATRIX_1:38; then A540: LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G* (i2,j2+1)) = {(F. k)/.len(F.k)} by A501,A511,A506,A513,A503,A539,GOBOARD7:18; F.(k+1) = (F.k)^ <*G*(i2,j2+1)*> by A429,A494,A496,A498,A499 ,A500,A501,A536,A539; hence thesis by A492,A495,A496,A540,SPPOL_2:30; end; suppose A541: i1 = i2+1 & j1 = j2; then [i2,j2-'1] in Indices G by A429,A494,A496,A498,A499,A500 ,A501,A536; then 1 <= j2-'1 by MATRIX_1:38; then A542: LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G*(i2,j2-'1)) = {(F. k)/.len(F.k)} by A501,A509,A512,A507,A505,A503,A541,GOBOARD7:15; F.(k+1) = (F.k)^ <*G*(i2,j2-'1)*> by A429,A494,A496,A498,A499 ,A500,A501,A536,A541; hence thesis by A492,A495,A496,A542,SPPOL_2:30; end; suppose A543: i1 = i2 & j1 = j2+1; then [i2+1,j2] in Indices G by A429,A494,A496,A498,A499,A500 ,A501,A536; then i2+1 <= len G by MATRIX_1:38; then A544: LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G* (i2+1,j2)) = {(F. k)/.len(F.k)} by A501,A511,A512,A504,A503,A543,GOBOARD7:17; F.(k+1) = (F.k)^ <*G*(i2+1,j2)*> by A429,A494,A496,A498,A499 ,A500,A501,A536,A543; hence thesis by A492,A495,A496,A544,SPPOL_2:30; end; end; hence thesis; end; end; hence thesis; end; end; now defpred P[Element of NAT] means F.$1 is one-to-one; assume A545: for k st k >= 1 holds for m st m in dom(F.k) & m <> len(F.k) holds (F.k)/.m <> (F.k)/.len(F.k); A546: for k st P[k] holds P[k+1] proof let k; assume A547: F.k is one-to-one; now let n,m such that A548: n in dom(F.(k+1)) and A549: m in dom(F.(k+1)) and A550: (F.(k+1))/.n = (F.(k+1))/.m; A551: n <= len(F.(k+1)) & m <= len(F.(k+1)) by A548,A549,FINSEQ_3:25; A552: 1 <= m by A549,FINSEQ_3:25; A553: 1 <= n by A548,FINSEQ_3:25; A554: ex i,j st [i,j] in Indices G & F.(k+1) = (F.k)^<*G*(i,j) *> by A464; A555: len(F.k) = k by A192; A556: len(F.(k+1)) = k+1 by A192; per cases by A551,A556,NAT_1:8; suppose A557: n <= k & m <= k; then A558: m in dom(F.k) by A552,A555,FINSEQ_3:25; then A559: (F.(k+1))/.m = (F.k)/.m by A554,FINSEQ_4:68; A560: n in dom(F.k) by A553,A555,A557,FINSEQ_3:25; then (F.(k+1))/.n = (F.k)/.n by A554,FINSEQ_4:68; hence n = m by A547,A550,A560,A558,A559,PARTFUN2:10; end; suppose n = k+1 & m <= k; hence n = m by A545,A549,A550,A556,NAT_1:12; end; suppose n <= k & m = k+1; hence n = m by A545,A548,A550,A556,NAT_1:12; end; suppose n = k+1 & m = k+1; hence n = m; end; end; hence thesis by PARTFUN2:9; end; A561: P[0] by A170; A562: for k holds P[k] from NAT_1:sch 1(A561,A546); A563: for k holds card rng(F.k) = k proof let k; F.k is one-to-one by A562; hence card rng(F.k) = len(F.k) by FINSEQ_4:62 .= k by A192; end; set k = (len G)*(width G)+1; F.k is_sequence_on G by A428; then card Values G <= (len G)*(width G) & card rng(F.k) <= card Values G by GOBRD13:8,MATRIX_1:46,NAT_1:43; then k > (len G)*(width G) & card rng(F.k) <= (len G)*(width G) by NAT_1:13,XXREAL_0:2; hence contradiction by A563; end; then A564: ex k be Nat st Q[k]; consider k be Nat such that A565: Q[k] and A566: for l be Nat st Q[l] holds k <= l from NAT_1:sch 5(A564); reconsider k as Element of NAT by ORDINAL1:def 12; consider m such that A567: m in dom(F.k) and A568: m <> len(F.k) and A569: (F.k)/.m = (F.k)/.len(F.k) by A565; A570: 1 <= m by A567,FINSEQ_3:25; reconsider f = F.k as non empty FinSequence of TOP-REAL 2 by A565; A571: f is_sequence_on G by A428; A572: m <= len f by A567,FINSEQ_3:25; then A573: m < len f by A568,XXREAL_0:1; then 1 < len f by A570,XXREAL_0:2; then A574: len f >= 1+1 by NAT_1:13; A575: P[0] by A170,CARD_1:27,SPPOL_2:26; for k holds P[k] from NAT_1:sch 1(A575,A491); then reconsider f as non constant special unfolded non empty FinSequence of TOP-REAL 2 by A571,A574,JORDAN8:4,5; A576: m+1 <= len f by A573,NAT_1:13; set g = f/^(m-'1); m-'1 <= m by NAT_D:44; then m-'1 < m+1 by NAT_1:13; then A577: m-'1 < len f by A576,XXREAL_0:2; then A578: len g = len f - (m-'1) by RFINSEQ:def 1; then (m-'1)-(m-'1) < len g by A577,XREAL_1:9; then reconsider g as non empty FinSequence of TOP-REAL 2 by CARD_1:27; len g in dom g by FINSEQ_5:6; then A579: g/.len g = f/.(m-'1+len g) by FINSEQ_5:27 .= f/.len f by A578; A580: len(F.k) = k by A192; A581: for i st 1 <= i & i < len g & 1 <= j & j < len g & g/.i = g/.j holds i = j proof let i such that A582: 1 <= i and A583: i < len g and A584: 1 <= j and A585: j < len g and A586: g/.i = g/.j and A587: i <> j; A588: i in dom g by A582,A583,FINSEQ_3:25; then A589: g/.i = f/.(m-'1+i) by FINSEQ_5:27; A590: j in dom g by A584,A585,FINSEQ_3:25; then A591: g/.j = f/.(m-'1+j) by FINSEQ_5:27; per cases by A587,XXREAL_0:1; suppose A592: i < j; set l = m-'1+j, m9= m-'1+i; A593: m9 < l by A592,XREAL_1:6; A594: len(F.l) = l by A192; A595: l < k by A580,A578,A585,XREAL_1:20; then A596: f|l = F.l by A490; 0+j <= l by XREAL_1:6; then A597: 1 <= l by A584,XXREAL_0:2; then l in dom(F.l) by A594,FINSEQ_3:25; then A598: (F.l)/.l = f/.l by A596,FINSEQ_4:70; 0+i <= m9 by XREAL_1:6; then 1 <= m9 by A582,XXREAL_0:2; then A599: m9 in dom(F.l) by A593,A594,FINSEQ_3:25; then (F.l)/.m9 = f/.m9 by A596,FINSEQ_4:70; hence contradiction by A566,A586,A589,A590,A593,A595,A597,A594,A599 ,A598,FINSEQ_5:27; end; suppose A600: j < i; set l = m-'1+i, m9= m-'1+j; A601: m9 < l by A600,XREAL_1:6; A602: len(F.l) = l by A192; A603: l < k by A580,A578,A583,XREAL_1:20; then A604: f|l = F.l by A490; 0+i <= l by XREAL_1:6; then A605: 1 <= l by A582,XXREAL_0:2; then l in dom(F.l) by A602,FINSEQ_3:25; then A606: (F.l)/.l = f/.l by A604,FINSEQ_4:70; 0+j <= m9 by XREAL_1:6; then 1 <= m9 by A584,XXREAL_0:2; then A607: m9 in dom(F.l) by A601,A602,FINSEQ_3:25; then (F.l)/.m9 = f/.m9 by A604,FINSEQ_4:70; hence contradiction by A566,A586,A588,A591,A601,A603,A605,A602,A607 ,A606,FINSEQ_5:27; end; end; A608: now consider i such that A609: 1 <= i & i+1 <= len G and A610: N-min C in cell(G,i,width G-'1) & N-min C <> G*(i,width G-'1) and A611: F.(1+1) = <*G*(i,width G),G*(i+1,width G)*> by A171; take i; thus 1 <= i & i+1 <= len G by A609; A612: f|2 = F.2 by A490,A580,A574; A613: len(f|2) = 2 by A574,FINSEQ_1:59; then 1 in dom(f|2) by FINSEQ_3:25; hence f/.1 = (f|2)/.1 by FINSEQ_4:70 .= G*(i,width G) by A611,A612,FINSEQ_4:17; 2 in dom(f|2) by A613,FINSEQ_3:25; hence f/.2 = (f|2)/.2 by FINSEQ_4:70 .= G*(i+1,width G) by A611,A612,FINSEQ_4:17; thus N-min C in cell(G,i,width G-'1) & N-min C <> G* (i,width G-'1) by A610; end; 1 in dom g by FINSEQ_5:6; then A614: g/.1 = f/.(m-'1+1) by FINSEQ_5:27 .= f/.m by A570,XREAL_1:235; A615: for i st 1 < i & i < j & j <= len g holds g/.i <> g/.j proof let i such that A616: 1 < i and A617: i < j and A618: j <= len g and A619: g/.i = g/.j; A620: 1 < j by A616,A617,XXREAL_0:2; A621: i < len g by A617,A618,XXREAL_0:2; then A622: 1 < len g by A616,XXREAL_0:2; per cases; suppose j <> len g; then j < len g by A618,XXREAL_0:1; hence contradiction by A581,A616,A617,A619,A620,A621; end; suppose j = len g; hence contradiction by A569,A614,A579,A581,A616,A617,A619,A622; end; end; m+1-(m-'1) <= len g by A576,A578,XREAL_1:9; then A623: m+1-(m-1) <= len g by A570,XREAL_1:233; then A624: 1+m-m+1 <= len g; A625: g is_sequence_on G by A428,JORDAN8:2; then A626: g is standard by JORDAN8:4; A627: g is non constant proof take 1,2; thus A628: 1 in dom g by FINSEQ_5:6; thus A629: 2 in dom g by A623,FINSEQ_3:25; then g/.1 <> g/.(1+1) by A626,FINSEQ_5:6,GOBOARD7:29; then g.1 <> g/.(1+1) by A628,PARTFUN1:def 6; hence thesis by A629,PARTFUN1:def 6; end; A630: for i st 1 <= i & i < j & j < len g holds g/.i <> g/.j proof let i such that A631: 1 <= i & i < j & j < len g and A632: g/.i = g/.j; 1 < j & i < len g by A631,XXREAL_0:2; hence contradiction by A581,A631,A632; end; g is s.c.c. proof let i,j such that A633: i+1 < j and A634: i > 1 & j < len g or j+1 < len g; A635: 1 < j by A633,NAT_1:12; A636: 1 <= i+1 by NAT_1:12; A637: j <= j+1 by NAT_1:12; then A638: i+1 < j+1 by A633,XXREAL_0:2; i < j by A633,NAT_1:13; then A639: i < j+1 by A637,XXREAL_0:2; per cases by A634,NAT_1:14; suppose A640: i > 1 & j < len g; then A641: j+1 <= len g by NAT_1:13; then A642: LSeg(g,j) = LSeg(g/.j,g/.(j+1)) by A635,TOPREAL1:def 3; consider i19,j19,i29,j29 being Element of NAT such that A643: [i19,j19] in Indices G and A644: g/.j = G*(i19,j19) and A645: [i29,j29] in Indices G and A646: g/.(j+1) = G*(i29,j29) and A647: i19 = i29 & j19+1 = j29 or i19+1 = i29 & j19 = j29 or i19 = i29+1 & j19 = j29 or i19 = i29 & j19 = j29+1 by A625,A635,A641,JORDAN8:3; A648: 1 <= i19 by A643,MATRIX_1:38; A649: j29 <= width G by A645,MATRIX_1:38; A650: 1 <= i29 by A645,MATRIX_1:38; A651: i19 <= len G by A643,MATRIX_1:38; A652: 1 <= j29 by A645,MATRIX_1:38; A653: j19 <= width G by A643,MATRIX_1:38; A654: i29 <= len G by A645,MATRIX_1:38; A655: 1 <= j19 by A643,MATRIX_1:38; A656: i+1 < len g by A633,A640,XXREAL_0:2; then A657: LSeg(g,i) = LSeg(g/.i,g/.(i+1)) by A640,TOPREAL1:def 3; A658: i < len g by A656,NAT_1:13; consider i1,j1,i2,j2 such that A659: [i1,j1] in Indices G and A660: g/.i = G*(i1,j1) and A661: [i2,j2] in Indices G and A662: g/.(i+1) = G*(i2,j2) and A663: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A625,A640,A656,JORDAN8:3; A664: 1 <= i1 by A659,MATRIX_1:38; A665: j2 <= width G by A661,MATRIX_1:38; A666: j1 <= width G by A659,MATRIX_1:38; A667: 1 <= j2 by A661,MATRIX_1:38; A668: 1 <= j1 by A659,MATRIX_1:38; A669: i2 <= len G by A661,MATRIX_1:38; A670: i1 <= len G by A659,MATRIX_1:38; A671: 1 < i+1 by A640,NAT_1:13; assume A672: LSeg(g,i) meets LSeg(g,j); A673: 1 <= i2 by A661,MATRIX_1:38; now per cases by A663,A647; suppose A674: i1 = i2 & j1+1 = j2 & i19 = i29 & j19+1 = j29; then A675: j1 = j19 or j1 = j19+1 or j1+1 = j19 by A657,A660,A662,A664,A670 ,A668,A665,A642,A644,A646,A648,A651,A655,A649,A672,GOBOARD7:22; i1 = i19 by A657,A660,A662,A664,A670,A668,A665,A642,A644,A646,A648 ,A651,A655,A649,A672,A674,GOBOARD7:19; hence contradiction by A581,A615,A633,A637,A636,A635,A639,A640,A656 ,A658,A660,A662,A641,A644,A646,A674,A675; end; suppose A676: i1 = i2 & j1+1 = j2 & i19+1 = i29 & j19 = j29; then i1 = i19 & j1 = j19 or i1 = i19 & j1+1 = j19 or i1 = i19+1 & j1 = j19 or i1 = i19+1 & j1+1 = j19 by A657,A660,A662,A664,A670,A668,A665 ,A642,A644,A646,A648,A655,A653,A654,A672,GOBOARD7:21; hence contradiction by A581,A615,A633,A638,A635,A639,A640,A671,A658 ,A660,A662,A641,A644,A646,A676; end; suppose A677: i1 = i2 & j1+1 = j2 & i19 = i29+1 & j19 = j29; then i1 = i29 & j19 = j1 or i1 = i29 & j1+1 = j19 or i1 = i29+1 & j19 = j1 or i1 = i29+1 & j1+1 = j19 by A657,A660,A662,A664,A670,A668,A665 ,A642,A644,A646,A651,A655,A653,A650,A672,GOBOARD7:21; hence contradiction by A581,A615,A633,A638,A635,A639,A640,A671,A658 ,A660,A662,A641,A644,A646,A677; end; suppose A678: i1 = i2 & j1+1 = j2 & i19 = i29 & j19 = j29+1; then A679: j1 = j29 or j1 = j29+1 or j1+1 = j29 by A657,A660,A662,A664,A670 ,A668,A665,A642,A644,A646,A648,A651,A653,A652,A672,GOBOARD7:22; i1 = i19 by A657,A660,A662,A664,A670,A668,A665,A642,A644,A646,A648 ,A651,A653,A652,A672,A678,GOBOARD7:19; hence contradiction by A581,A615,A633,A638,A635,A640,A671,A658,A660 ,A662,A641,A644,A646,A678,A679; end; suppose A680: i1+1 = i2 & j1 = j2 & i19 = i29 & j19+1 = j29; then i19 = i1 & j1 = j19 or i19 = i1 & j19+1 = j1 or i19 = i1+1 & j1 = j19 or i19 = i1+1 & j19+1 = j1 by A657,A660,A662,A664,A668,A666,A669 ,A642,A644,A646,A648,A651,A655,A649,A672,GOBOARD7:21; hence contradiction by A581,A615,A633,A638,A635,A639,A640,A671,A658 ,A660,A662,A641,A644,A646,A680; end; suppose A681: i1+1 = i2 & j1 = j2 & i19+1 = i29 & j19 = j29; then A682: i1 = i19 or i1 = i19+1 or i1+1 = i19 by A657,A660,A662,A664,A668 ,A666,A669,A642,A644,A646,A648,A655,A653,A654,A672,GOBOARD7:23; j1 = j19 by A657,A660,A662,A664,A668,A666,A669,A642,A644,A646,A648 ,A655,A653,A654,A672,A681,GOBOARD7:20; hence contradiction by A581,A615,A633,A637,A636,A635,A639,A640,A656 ,A658,A660,A662,A641,A644,A646,A681,A682; end; suppose A683: i1+1 = i2 & j1 = j2 & i19 = i29+1 & j19 = j29; then A684: i1 = i29 or i1 = i29+1 or i1+1 = i29 by A657,A660,A662,A664,A668 ,A666,A669,A642,A644,A646,A651,A655,A653,A650,A672,GOBOARD7:23; j1 = j19 by A657,A660,A662,A664,A668,A666,A669,A642,A644,A646,A651 ,A655,A653,A650,A672,A683,GOBOARD7:20; hence contradiction by A581,A615,A638,A635,A639,A640,A671,A658,A660 ,A662,A641,A644,A646,A683,A684; end; suppose A685: i1+1 = i2 & j1 = j2 & i19 = i29 & j19 = j29+1; then i19 = i1 & j1 = j29 or i19 = i1 & j29+1 = j1 or i19 = i1+1 & j1 = j29 or i19 = i1+1 & j29+1 = j1 by A657,A660,A662,A664,A668,A666,A669 ,A642,A644,A646,A648,A651,A653,A652,A672,GOBOARD7:21; hence contradiction by A581,A615,A633,A638,A635,A639,A640,A671,A658 ,A660,A662,A641,A644,A646,A685; end; suppose A686: i1 = i2+1 & j1 = j2 & i19 = i29 & j19+1 = j29; then i19 = i2 & j19 = j1 or i19 = i2 & j19+1 = j1 or i19 = i2+1 & j19 = j1 or i19 = i2+1 & j19+1 = j1 by A657,A660,A662,A670,A668,A666,A673 ,A642,A644,A646,A648,A651,A655,A649,A672,GOBOARD7:21; hence contradiction by A581,A615,A633,A638,A635,A639,A640,A671,A658 ,A660,A662,A641,A644,A646,A686; end; suppose A687: i1 = i2+1 & j1 = j2 & i19+1 = i29 & j19 = j29; then A688: i2 = i19 or i2 = i19+1 or i2+1 = i19 by A657,A660,A662,A670,A668 ,A666,A673,A642,A644,A646,A648,A655,A653,A654,A672,GOBOARD7:23; j1 = j19 by A657,A660,A662,A670,A668,A666,A673,A642,A644,A646,A648 ,A655,A653,A654,A672,A687,GOBOARD7:20; hence contradiction by A581,A615,A633,A638,A635,A640,A671,A658,A660 ,A662,A641,A644,A646,A687,A688; end; suppose A689: i1 = i2+1 & j1 = j2 & i19 = i29+1 & j19 = j29; then A690: i2 = i29 or i2 = i29+1 or i2+1 = i29 by A657,A660,A662,A670,A668 ,A666,A673,A642,A644,A646,A651,A655,A653,A650,A672,GOBOARD7:23; j1 = j19 by A657,A660,A662,A670,A668,A666,A673,A642,A644,A646,A651 ,A655,A653,A650,A672,A689,GOBOARD7:20; hence contradiction by A615,A633,A638,A639,A640,A671,A660,A662,A641 ,A644,A646,A689,A690; end; suppose A691: i1 = i2+1 & j1 = j2 & i19 = i29 & j19 = j29+1; then i19 = i2 & j29 = j1 or i19 = i2 & j29+1 = j1 or i19 = i2+1 & j29 = j1 or i19 = i2+1 & j29+1 = j1 by A657,A660,A662,A670,A668,A666,A673 ,A642,A644,A646,A648,A651,A653,A652,A672,GOBOARD7:21; hence contradiction by A581,A615,A633,A638,A635,A639,A640,A671,A658 ,A660,A662,A641,A644,A646,A691; end; suppose A692: i1 = i2 & j1 = j2+1 & i19 = i29 & j19+1 = j29; then A693: j2 = j19 or j2 = j19+1 or j2+1 = j19 by A657,A660,A662,A664,A670 ,A666,A667,A642,A644,A646,A648,A651,A655,A649,A672,GOBOARD7:22; i1 = i19 by A657,A660,A662,A664,A670,A666,A667,A642,A644,A646,A648 ,A651,A655,A649,A672,A692,GOBOARD7:19; hence contradiction by A581,A615,A633,A638,A635,A640,A671,A658,A660 ,A662,A641,A644,A646,A692,A693; end; suppose A694: i1 = i2 & j1 = j2+1 & i19+1 = i29 & j19 = j29; then i1 = i19 & j2 = j19 or i1 = i19 & j2+1 = j19 or i1 = i19+1 & j2 = j19 or i1 = i19+1 & j2+1 = j19 by A657,A660,A662,A664,A670,A666,A667 ,A642,A644,A646,A648,A655,A653,A654,A672,GOBOARD7:21; hence contradiction by A581,A615,A633,A638,A635,A639,A640,A671,A658 ,A660,A662,A641,A644,A646,A694; end; suppose A695: i1 = i2 & j1 = j2+1 & i19 = i29+1 & j19 = j29; then i1 = i29 & j2 = j19 or i1 = i29 & j2+1 = j19 or i1 = i29+1 & j2 = j19 or i1 = i29+1 & j2+1 = j19 by A657,A660,A662,A664,A670,A666,A667 ,A642,A644,A646,A651,A655,A653,A650,A672,GOBOARD7:21; hence contradiction by A581,A615,A633,A638,A635,A639,A640,A671,A658 ,A660,A662,A641,A644,A646,A695; end; suppose A696: i1 = i2 & j1 = j2+1 & i19 = i29 & j19 = j29+1; then A697: j2 = j29 or j2 = j29+1 or j2+1 = j29 by A657,A660,A662,A664,A670 ,A666,A667,A642,A644,A646,A648,A651,A653,A652,A672,GOBOARD7:22; i1 = i19 by A657,A660,A662,A664,A670,A666,A667,A642,A644,A646,A648 ,A651,A653,A652,A672,A696,GOBOARD7:19; hence contradiction by A615,A633,A638,A639,A640,A671,A660,A662,A641 ,A644,A646,A696,A697; end; end; hence contradiction; end; suppose i = 0 & j+1 < len g; then LSeg(g,i) = {} by TOPREAL1:def 3; hence thesis by XBOOLE_1:65; end; suppose A698: 1 <= i & j+1 < len g; then A699: i+1 < len g by A638,XXREAL_0:2; then A700: LSeg(g,i) = LSeg(g/.i,g/.(i+1)) by A698,TOPREAL1:def 3; A701: i < len g by A699,NAT_1:13; consider i1,j1,i2,j2 such that A702: [i1,j1] in Indices G and A703: g/.i = G*(i1,j1) and A704: [i2,j2] in Indices G and A705: g/.(i+1) = G*(i2,j2) and A706: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A625,A698,A699,JORDAN8:3; A707: 1 <= i1 by A702,MATRIX_1:38; A708: j2 <= width G by A704,MATRIX_1:38; A709: j1 <= width G by A702,MATRIX_1:38; A710: 1 <= j2 by A704,MATRIX_1:38; A711: 1 <= j1 by A702,MATRIX_1:38; A712: i2 <= len G by A704,MATRIX_1:38; A713: i1 <= len G by A702,MATRIX_1:38; A714: 1 < i+1 by A698,NAT_1:13; assume A715: LSeg(g,i) meets LSeg(g,j); consider i19,j19,i29,j29 being Element of NAT such that A716: [i19,j19] in Indices G and A717: g/.j = G*(i19,j19) and A718: [i29,j29] in Indices G and A719: g/.(j+1) = G*(i29,j29) and A720: i19 = i29 & j19+1 = j29 or i19+1 = i29 & j19 = j29 or i19 = i29+1 & j19 = j29 or i19 = i29 & j19 = j29+1 by A625,A635,A698,JORDAN8:3; A721: 1 <= i19 by A716,MATRIX_1:38; A722: j29 <= width G by A718,MATRIX_1:38; A723: 1 <= i29 by A718,MATRIX_1:38; A724: i19 <= len G by A716,MATRIX_1:38; A725: 1 <= j29 by A718,MATRIX_1:38; A726: j19 <= width G by A716,MATRIX_1:38; A727: i29 <= len G by A718,MATRIX_1:38; A728: 1 <= j19 by A716,MATRIX_1:38; A729: j < len g by A698,NAT_1:12; A730: LSeg(g,j) = LSeg(g/.j,g/.(j+1)) by A635,A698,TOPREAL1:def 3; A731: 1 <= i2 by A704,MATRIX_1:38; now per cases by A706,A720; suppose A732: i1 = i2 & j1+1 = j2 & i19 = i29 & j19+1 = j29; then A733: j1 = j19 or j1 = j19+1 or j1+1 = j19 by A700,A703,A705,A707,A713 ,A711,A708,A730,A717,A719,A721,A724,A728,A722,A715,GOBOARD7:22; i1 = i19 by A700,A703,A705,A707,A713,A711,A708,A730,A717,A719,A721 ,A724,A728,A722,A715,A732,GOBOARD7:19; hence contradiction by A630,A633,A638,A636,A639,A698,A729,A703,A705 ,A717,A719,A732,A733; end; suppose A734: i1 = i2 & j1+1 = j2 & i19+1 = i29 & j19 = j29; then i1 = i19 & j1 = j19 or i1 = i19 & j1+1 = j19 or i1 = i19+1 & j1 = j19 or i1 = i19+1 & j1+1 = j19 by A700,A703,A705,A707,A713,A711,A708 ,A730,A717,A719,A721,A728,A726,A727,A715,GOBOARD7:21; hence contradiction by A581,A630,A633,A638,A635,A639,A698,A714,A701 ,A729,A703,A705,A717,A719,A734; end; suppose A735: i1 = i2 & j1+1 = j2 & i19 = i29+1 & j19 = j29; then i1 = i29 & j19 = j1 or i1 = i29 & j1+1 = j19 or i1 = i29+1 & j19 = j1 or i1 = i29+1 & j1+1 = j19 by A700,A703,A705,A707,A713,A711,A708 ,A730,A717,A719,A724,A728,A726,A723,A715,GOBOARD7:21; hence contradiction by A581,A630,A633,A638,A635,A639,A698,A714,A701 ,A729,A703,A705,A717,A719,A735; end; suppose A736: i1 = i2 & j1+1 = j2 & i19 = i29 & j19 = j29+1; then A737: j1 = j29 or j1 = j29+1 or j1+1 = j29 by A700,A703,A705,A707,A713 ,A711,A708,A730,A717,A719,A721,A724,A726,A725,A715,GOBOARD7:22; i1 = i19 by A700,A703,A705,A707,A713,A711,A708,A730,A717,A719,A721 ,A724,A726,A725,A715,A736,GOBOARD7:19; hence contradiction by A581,A615,A633,A638,A635,A698,A714,A701,A729 ,A703,A705,A717,A719,A736,A737; end; suppose A738: i1+1 = i2 & j1 = j2 & i19 = i29 & j19+1 = j29; then i19 = i1 & j1 = j19 or i19 = i1 & j19+1 = j1 or i19 = i1+1 & j1 = j19 or i19 = i1+1 & j19+1 = j1 by A700,A703,A705,A707,A711,A709,A712 ,A730,A717,A719,A721,A724,A728,A722,A715,GOBOARD7:21; hence contradiction by A581,A630,A633,A638,A636,A635,A639,A698,A701 ,A729,A703,A705,A717,A719,A738; end; suppose A739: i1+1 = i2 & j1 = j2 & i19+1 = i29 & j19 = j29; then A740: i1 = i19 or i1 = i19+1 or i1+1 = i19 by A700,A703,A705,A707,A711 ,A709,A712,A730,A717,A719,A721,A728,A726,A727,A715,GOBOARD7:23; j1 = j19 by A700,A703,A705,A707,A711,A709,A712,A730,A717,A719,A721 ,A728,A726,A727,A715,A739,GOBOARD7:20; hence contradiction by A630,A633,A638,A636,A639,A698,A729,A703,A705 ,A717,A719,A739,A740; end; suppose A741: i1+1 = i2 & j1 = j2 & i19 = i29+1 & j19 = j29; then A742: i1 = i29 or i1 = i29+1 or i1+1 = i29 by A700,A703,A705,A707,A711 ,A709,A712,A730,A717,A719,A724,A728,A726,A723,A715,GOBOARD7:23; j1 = j19 by A700,A703,A705,A707,A711,A709,A712,A730,A717,A719,A724 ,A728,A726,A723,A715,A741,GOBOARD7:20; hence contradiction by A581,A630,A638,A635,A639,A698,A714,A701,A729 ,A703,A705,A717,A719,A741,A742; end; suppose A743: i1+1 = i2 & j1 = j2 & i19 = i29 & j19 = j29+1; then i19 = i1 & j1 = j29 or i19 = i1 & j29+1 = j1 or i19 = i1+1 & j1 = j29 or i19 = i1+1 & j29+1 = j1 by A700,A703,A705,A707,A711,A709,A712 ,A730,A717,A719,A721,A724,A726,A725,A715,GOBOARD7:21; hence contradiction by A581,A630,A633,A638,A635,A639,A698,A714,A701 ,A729,A703,A705,A717,A719,A743; end; suppose A744: i1 = i2+1 & j1 = j2 & i19 = i29 & j19+1 = j29; then i19 = i2 & j19 = j1 or i19 = i2 & j19+1 = j1 or i19 = i2+1 & j19 = j1 or i19 = i2+1 & j19+1 = j1 by A700,A703,A705,A713,A711,A709,A731 ,A730,A717,A719,A721,A724,A728,A722,A715,GOBOARD7:21; hence contradiction by A581,A630,A633,A638,A635,A639,A698,A714,A701 ,A729,A703,A705,A717,A719,A744; end; suppose A745: i1 = i2+1 & j1 = j2 & i19+1 = i29 & j19 = j29; then A746: i2 = i19 or i2 = i19+1 or i2+1 = i19 by A700,A703,A705,A713,A711 ,A709,A731,A730,A717,A719,A721,A728,A726,A727,A715,GOBOARD7:23; j1 = j19 by A700,A703,A705,A713,A711,A709,A731,A730,A717,A719,A721 ,A728,A726,A727,A715,A745,GOBOARD7:20; hence contradiction by A581,A615,A633,A638,A635,A698,A714,A701,A729 ,A703,A705,A717,A719,A745,A746; end; suppose A747: i1 = i2+1 & j1 = j2 & i19 = i29+1 & j19 = j29; then A748: i2 = i29 or i2 = i29+1 or i2+1 = i29 by A700,A703,A705,A713,A711 ,A709,A731,A730,A717,A719,A724,A728,A726,A723,A715,GOBOARD7:23; j1 = j19 by A700,A703,A705,A713,A711,A709,A731,A730,A717,A719,A724 ,A728,A726,A723,A715,A747,GOBOARD7:20; hence contradiction by A630,A633,A638,A639,A698,A714,A729,A703,A705 ,A717,A719,A747,A748; end; suppose A749: i1 = i2+1 & j1 = j2 & i19 = i29 & j19 = j29+1; then i19 = i2 & j29 = j1 or i19 = i2 & j29+1 = j1 or i19 = i2+1 & j29 = j1 or i19 = i2+1 & j29+1 = j1 by A700,A703,A705,A713,A711,A709,A731 ,A730,A717,A719,A721,A724,A726,A725,A715,GOBOARD7:21; hence contradiction by A581,A630,A633,A638,A635,A639,A698,A714,A701 ,A729,A703,A705,A717,A719,A749; end; suppose A750: i1 = i2 & j1 = j2+1 & i19 = i29 & j19+1 = j29; then A751: j2 = j19 or j2 = j19+1 or j2+1 = j19 by A700,A703,A705,A707,A713 ,A709,A710,A730,A717,A719,A721,A724,A728,A722,A715,GOBOARD7:22; i1 = i19 by A700,A703,A705,A707,A713,A709,A710,A730,A717,A719,A721 ,A724,A728,A722,A715,A750,GOBOARD7:19; hence contradiction by A581,A615,A633,A638,A635,A698,A714,A701,A729 ,A703,A705,A717,A719,A750,A751; end; suppose A752: i1 = i2 & j1 = j2+1 & i19+1 = i29 & j19 = j29; then i1 = i19 & j2 = j19 or i1 = i19 & j2+1 = j19 or i1 = i19+1 & j2 = j19 or i1 = i19+1 & j2+1 = j19 by A700,A703,A705,A707,A713,A709,A710 ,A730,A717,A719,A721,A728,A726,A727,A715,GOBOARD7:21; hence contradiction by A581,A630,A633,A638,A635,A639,A698,A714,A701 ,A729,A703,A705,A717,A719,A752; end; suppose A753: i1 = i2 & j1 = j2+1 & i19 = i29+1 & j19 = j29; then i1 = i29 & j2 = j19 or i1 = i29 & j2+1 = j19 or i1 = i29+1 & j2 = j19 or i1 = i29+1 & j2+1 = j19 by A700,A703,A705,A707,A713,A709,A710 ,A730,A717,A719,A724,A728,A726,A723,A715,GOBOARD7:21; hence contradiction by A581,A630,A633,A638,A635,A639,A698,A714,A701 ,A729,A703,A705,A717,A719,A753; end; suppose A754: i1 = i2 & j1 = j2+1 & i19 = i29 & j19 = j29+1; then A755: j2 = j29 or j2 = j29+1 or j2+1 = j29 by A700,A703,A705,A707,A713 ,A709,A710,A730,A717,A719,A721,A724,A726,A725,A715,GOBOARD7:22; i1 = i19 by A700,A703,A705,A707,A713,A709,A710,A730,A717,A719,A721 ,A724,A726,A725,A715,A754,GOBOARD7:19; hence contradiction by A630,A633,A638,A639,A698,A714,A729,A703,A705 ,A717,A719,A754,A755; end; end; hence contradiction; end; end; then reconsider g as standard non constant special_circular_sequence by A569,A614,A579,A625 ,A627,FINSEQ_6:def 1,JORDAN8:4; A756: for i st 1 <= i & i+1 <= len f holds right_cell(f,i,G) = Cl Int right_cell(f,i,G) proof let i such that A757: 1 <= i & i+1 <= len f; consider i1,j1,i2,j2 such that A758: [i1,j1] in Indices G and A759: f/.i = G*(i1,j1) and A760: [i2,j2] in Indices G and A761: f/.(i+1) = G*(i2,j2) and A762: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A571,A757,JORDAN8:3; A763: i1 <= len G by A758,MATRIX_1:38; A764: j2 <= width G by A760,MATRIX_1:38; A765: j1 <= width G by A758,MATRIX_1:38; A766: j1+1 > j1 & j2+1 > j2 by NAT_1:13; A767: i2 <= len G by A760,MATRIX_1:38; A768: i1+1 > i1 & i2+1 > i2 by NAT_1:13; per cases by A762; suppose i1 = i2 & j1+1 = j2; then right_cell(f,i,G) = cell(G,i1,j1) by A571,A757,A758,A759,A760,A761 ,A766,GOBRD13:def 2; hence thesis by A763,A765,GOBRD11:35; end; suppose A769: i1+1 = i2 & j1 = j2; A770: j1-'1 <= width G by A765,NAT_D:44; right_cell(f,i,G) = cell(G,i1,j1-'1) by A571,A757,A758,A759,A760,A761 ,A768,A769,GOBRD13:def 2; hence thesis by A763,A770,GOBRD11:35; end; suppose i1 = i2+1 & j1 = j2; then right_cell(f,i,G) = cell(G,i2,j2) by A571,A757,A758,A759,A760,A761 ,A768,GOBRD13:def 2; hence thesis by A767,A764,GOBRD11:35; end; suppose A771: i1 = i2 & j1 = j2+1; A772: i1-'1 <= len G by A763,NAT_D:44; right_cell(f,i,G) = cell(G,i1-'1,j2) by A571,A757,A758,A759,A760,A761 ,A766,A771,GOBRD13:def 2; hence thesis by A764,A772,GOBRD11:35; end; end; now A773: for h being standard non constant special_circular_sequence st L~ h c= L~f for Comp being Subset of TOP-REAL 2 st Comp is_a_component_of (L~h)` for n st 1 <= n & n+1 <= len f & f/.n in Comp & not f/.n in L~h holds C meets Comp proof let h be standard non constant special_circular_sequence such that A774: L~h c= L~f; let Comp be Subset of TOP-REAL 2 such that A775: Comp is_a_component_of (L~h)`; let n such that A776: 1 <= n & n+1 <= len f and A777: f/.n in Comp and A778: not f/.n in L~h; reconsider rc = right_cell(f,n,G)\L~h as Subset of TOP-REAL 2; f/.n in right_cell(f,n,G) by A571,A776,Th8; then f/.n in rc by A778,XBOOLE_0:def 5; then A779: rc meets Comp by A777,XBOOLE_0:3; A780: rc meets C proof right_cell(f,n,G) meets C by A428,A776; then consider p being set such that A781: p in right_cell(f,n,G) and A782: p in C by XBOOLE_0:3; reconsider p as Element of TOP-REAL 2 by A781; now take a = p; now assume p in L~h; then consider j such that A783: 1 <= j & j+1 <= len f and A784: p in LSeg(f,j) by A774,SPPOL_2:13; p in left_cell(f,j,G) /\ right_cell(f,j,G) by A428,A783,A784, GOBRD13:29; then A785: p in left_cell(f,j,G) by XBOOLE_0:def 4; left_cell(f,j,G) misses C by A428,A783; hence contradiction by A782,A785,XBOOLE_0:3; end; hence a in rc by A781,XBOOLE_0:def 5; thus a in C by A782; end; hence thesis by XBOOLE_0:3; end; Int right_cell(f,n,G) misses L~f by A571,A776,Th15; then Int right_cell(f,n,G) misses L~h by A774,XBOOLE_1:63; then A786: Int right_cell(f,n,G) c= (L~h)` by SUBSET_1:23; A787: rc = right_cell(f,n,G) /\ (L~h)` by SUBSET_1:13; then A788: rc c= (L~h)` by XBOOLE_1:17; rc c= right_cell(f,n,G) by XBOOLE_1:36; then A789: Int right_cell(f,n,G) c= right_cell(f,n,G) & rc c= Cl Int right_cell(f,n,G) by A571,A776,Th11,TOPS_1:16; Int right_cell(f,n,G) is convex by A571,A776,Th10; then rc is connected by A787,A786,A789,CONNSP_1:18,XBOOLE_1:19; then rc c= Comp by A775,A779,A788,GOBOARD9:4; hence thesis by A780,XBOOLE_1:63; end; (L~g)` is open by TOPS_1:3; then A790: (L~g)` = Int (L~g)` by TOPS_1:23; A791: L~g c= L~f by JORDAN3:40; A792: for j,k st 1 <= j & j <= k holds (F.k)/.j = (F.j)/.j proof let j,k; assume that A793: 1 <= j and A794: j <= k; j <= len(F.k) by A192,A794; then len(F.k|j) = j by FINSEQ_1:59; then A795: j in dom((F.k)|j) by A793,FINSEQ_3:25; (F.k)|j = F.j by A490,A794; hence thesis by A795,FINSEQ_4:70; end; assume m <> 1; then A796: 1 < m by A570,XXREAL_0:1; A797: for n st 1 <= n & n <= m-'1 holds not f/.n in L~g proof A798: 2 <= len G by A231,NAT_1:12; let n such that A799: 1 <= n and A800: n <= m-'1; set p = f/.n; A801: n <= len f by A577,A800,XXREAL_0:2; then A802: p in Values G by A428,A799,Th6; assume p in L~g; then consider j such that A803: m-'1+1 <= j and A804: j+1 <= len f and A805: p in LSeg(f,j) by A577,Th7; A806: j < k by A580,A804,NAT_1:13; A807: n < m-'1+1 by A800,NAT_1:13; then A808: n < j by A803,XXREAL_0:2; A809: m-'1+1=m by A570,XREAL_1:235; then A810: 1 < j by A796,A803,XXREAL_0:2; per cases by A2,A428,A804,A805,A810,A798,A802,Th23; suppose A811: p = f/.j; A812: n <> len(F.j) by A192,A803,A807; n <= len(F.j) by A192,A808; then A813: n in dom(F.j) by A799,FINSEQ_3:25; (F.j)/.n = (F.n)/.n by A792,A799,A808 .= p by A580,A792,A799,A801 .= (F.j)/.j by A792,A810,A806,A811 .= (F.j)/.len(F.j) by A192; hence contradiction by A566,A810,A806,A813,A812; end; suppose A814: p = f/.(j+1); now per cases by A580,A804,XXREAL_0:1; suppose A815: j+1 = k; A816: n <> len(F.m) by A192,A807,A809; n <= len(F.m) by A192,A807,A809; then A817: n in dom(F.m) by A799,FINSEQ_3:25; (F.m)/.n = (F.n)/.n by A792,A799,A807,A809 .= (F.k)/.k by A580,A792,A799,A801,A814,A815 .= (F.m)/.m by A569,A580,A570,A572,A792 .= (F.m)/.len(F.m) by A192; hence contradiction by A566,A580,A570,A573,A817,A816; end; suppose A818: j+1 < k; set l = j+1; A819: 1 <= l by NAT_1:11; A820: n < n+1 & n+1 < l by A808,XREAL_1:6,29; then A821: n <> len(F.l) by A192; A822: n < l by A820,XXREAL_0:2; then n <= len(F.l) by A192; then A823: n in dom(F.l) by A799,FINSEQ_3:25; (F.l)/.n = (F.n)/.n by A792,A799,A822 .= p by A580,A792,A799,A801 .= (F.l)/.l by A792,A814,A818,A819 .= (F.l)/.len(F.l) by A192; hence contradiction by A566,A818,A823,A821,NAT_1:11; end; end; hence contradiction; end; end; C meets RightComp Rev g proof 1 <= len g by A624,XREAL_1:145; then A824: len g-'1+2 = len g+1 by Lm1; A825: 1 - 1 < m - 1 by A796,XREAL_1:9; A826: m-'1+2 = m+1 by A570,Lm1; set l = (m-'1)+(len g-'1); set a = f/.(m-'1); set rg=Rev g; set p = rg/.1, q = rg/.2; A827: 1+1 - 1 <= len g - 1 by A623,XREAL_1:9; 1+1-'1 <= len g-'1 by A623,NAT_D:42; then A828: 1 <= len g-'1 by NAT_D:34; then (m-'1)+1 <= l by XREAL_1:6; then m-'1 < l by NAT_1:13; then A829: m-'1 <> len(F.l) by A192; A830: 1+1 <= len rg by A623,FINSEQ_5:def 3; then 1+1-'1 <= len rg-'1 by NAT_D:42; then A831: 1 <= len rg -'1 by NAT_D:34; 1 < len rg by A830,NAT_1:13; then A832: len rg -'1+1 = len rg by XREAL_1:235; A833: rg is_sequence_on G by A625,Th5; then consider p1,p2,q1,q2 being Element of NAT such that A834: [p1,p2] in Indices G and A835: p = G*(p1,p2) and A836: [q1,q2] in Indices G and A837: q = G*(q1,q2) and A838: p1 = q1 & p2+1 = q2 or p1+1 = q1 & p2 = q2 or p1 = q1+1 & p2 = q2 or p1 = q1 & p2 = q2+1 by A830,JORDAN8:3; A839: 1 <= p1 by A834,MATRIX_1:38; A840: p2 <= width G by A834,MATRIX_1:38; A841: p1 <= len G by A834,MATRIX_1:38; A842: 1 <= p2 by A834,MATRIX_1:38; A843: p = f/.m by A569,A579,FINSEQ_5:65; len g-'1 <= len g by NAT_D:44; then A844: len g-'1 in dom g by A828,FINSEQ_3:25; then A845: q = g/.(len g-'1) by A824,FINSEQ_5:66 .= f/.l by A844,FINSEQ_5:27; l = m+(len g -'1)-'1 by A570,NAT_D:38 .= (len g -'1)+m - 1 by A828,NAT_D:37 .= (len g - 1)+m - 1 by A827,XREAL_0:def 2 .= ((k - (m - 1)) - 1)+m - 1 by A580,A578,A825,XREAL_0:def 2 .= k - 1; then A846: k = l+1; then A847: l < k by XREAL_1:29; len g-'1 <= l by NAT_1:11; then A848: 1 <= l by A828,XXREAL_0:2; then A849: right_cell(f,l,G) meets C by A428,A580,A846; A850: m-'1+1 = m by A570,XREAL_1:235; then A851: 1 <= m-'1 by A796,NAT_1:13; then A852: right_cell(f,m-'1,G) meets C by A428,A572,A850; m-'1 <= l by NAT_1:11; then m-'1 <= len(F.l) by A192; then A853: m-'1 in dom(F.l) by A851,FINSEQ_3:25; not a in L~g by A797,A851; then A854: not a in L~rg by SPPOL_2:22; per cases by A838; suppose A855: p1 = q1 & p2+1 = q2; consider a1,a2,p91,p92 being Element of NAT such that A856: [a1,a2] in Indices G and A857: a = G*(a1,a2) and A858: [p91,p92] in Indices G & p = G*(p91,p92) and A859: a1 = p91 & a2+1 = p92 or a1+1 = p91 & a2 = p92 or a1 = p91+1 & a2 = p92 or a1 = p91 & a2 = p92+1 by A571,A572,A843,A850,A851,JORDAN8:3 ; A860: 1 <= a2 by A856,MATRIX_1:38; thus thesis proof per cases by A859; suppose A861: a1 = p91 & a2+1 = p92; A862: m-'1+1 <= len (F.m) & f/.(m-'1+1) = (F.m)/.m by A192,A580,A570 ,A572,A792,A850; A863: F.k|(m+1)=F.(m+1) by A490,A580,A576; A864: a1 = p1 by A834,A835,A858,A861,GOBOARD1:5; A865: m-'1 <= m by A850,NAT_1:11; A866: f/.(m-'1) = (F.(m-'1))/.(m-'1) by A580,A577,A792,A851 .= (F.m)/.(m-'1) by A792,A851,A865; A867: 2 in dom g by A623,FINSEQ_3:25; A868: a2+1 = p2 by A834,A835,A858,A861,GOBOARD1:5; then A869: p2-'1 = a2 by NAT_D:34; right_cell(f,l,G) = cell(G,p1-'1,p2) by A428,A569,A580,A843,A846 ,A848,A845,A834,A835,A836,A837,A855,GOBRD13:28 .= front_left_cell(F.m,m-'1,G) by A428,A843,A850,A851,A834,A835 ,A856,A857,A864,A868,A866,A862,GOBRD13:34; then F.(m+1) turns_left m-'1,G by A458,A796,A849; then A870: a2+1 > a2 & f turns_left m-'1,G by A851,A826,A863,GOBRD13:44 ,NAT_1:13; len rg-'1+2 = len g +1 by A824,FINSEQ_5:def 3; then A871: rg/.(len rg-'1) = g/.2 by A867,FINSEQ_5:66 .= f/.(m+1) by A826,A867,FINSEQ_5:27; A872: p = g/.1 by A569,A614,A579,FINSEQ_5:65 .= rg/.len g by FINSEQ_5:65 .= rg/.len rg by FINSEQ_5:def 3; set rc = right_cell(rg,len rg-'1,G)\L~rg; A873: RightComp rg is_a_component_of (L~rg)` by GOBOARD9:def 2; A874: p1-'1+1 = p1 by A839,XREAL_1:235; A875: p2+1 > a2+1 by A868,NAT_1:13; then A876: [p1-'1,p2] in Indices G by A843,A850,A834,A835,A856,A857,A870, GOBRD13:def 7; then A877: 1 <= p1-'1 by MATRIX_1:38; f/.(m+1) = G*(p1-'1,p2) by A843,A850,A826,A834,A835,A856,A857 ,A875,A870,GOBRD13:def 7; then right_cell(rg,len rg-'1,G) = cell(G,p1-'1,a2) by A833,A831 ,A832,A834,A835,A876,A869,A874,A871,A872,GOBRD13:24; then a in right_cell(rg,len rg-'1,G) by A841,A840,A857,A860,A864 ,A868,A877,A874,Th20; then A878: a in rc by A854,XBOOLE_0:def 5; rc c= RightComp rg & L~rg c= L~f by A791,A833,A831,A832,Th27, SPPOL_2:22; hence thesis by A572,A773,A850,A851,A854,A878,A873; end; suppose A879: a1+1 = p91 & a2 = p92; then a1+1 = p1 by A834,A835,A858,GOBOARD1:5; then A880: q1-'1 = a1 by A855,NAT_D:34; a2 = p2 by A834,A835,A858,A879,GOBOARD1:5; then right_cell(f,l,G) = cell(G,a1,a2) by A428,A569,A580,A843 ,A846,A848,A845,A834,A835,A836,A837,A855,A880,GOBRD13:28 .= left_cell(f,m-'1,G) by A428,A572,A843,A850,A851,A856,A857 ,A858,A879,GOBRD13:23; hence thesis by A428,A572,A850,A851,A849; end; suppose a1 = p91+1 & a2 = p92; then a1 = p1+1 & a2 = p2 by A834,A835,A858,GOBOARD1:5; then right_cell(f,m-'1,G) = cell(G,p1,p2) by A428,A572,A843,A850 ,A851,A834,A835,A856,A857,GOBRD13:26 .= left_cell(f,l,G) by A428,A569,A580,A843,A846,A848,A845,A834 ,A835,A836,A837,A855,GOBRD13:27; hence thesis by A428,A580,A846,A848,A852; end; suppose a1 = p91 & a2 = p92+1; then A881: a1 = q1 & a2 = q2 by A834,A835,A855,A858,GOBOARD1:5; (F.l)/.(m-'1) = (F.(m-'1))/.(m-'1) by A792,A851,NAT_1:11 .= q by A580,A577,A792,A851,A837,A857,A881 .= (F.l)/.l by A792,A847,A848,A845 .= (F.l)/.len(F.l) by A192; hence thesis by A566,A847,A848,A853,A829; end; end; end; suppose A882: p1+1 = q1 & p2 = q2; consider a1,a2,p91,p92 being Element of NAT such that A883: [a1,a2] in Indices G and A884: a = G*(a1,a2) and A885: [p91,p92] in Indices G & p = G*(p91,p92) and A886: a1 = p91 & a2+1 = p92 or a1+1 = p91 & a2 = p92 or a1 = p91+1 & a2 = p92 or a1 = p91 & a2 = p92+1 by A571,A572,A843,A850,A851,JORDAN8:3 ; A887: 1 <= a1 by A883,MATRIX_1:38; thus thesis proof per cases by A886; suppose A888: a1 = p91 & a2+1 = p92; then a2+1 = p2 by A834,A835,A885,GOBOARD1:5; then A889: q2-'1 = a2 by A882,NAT_D:34; A890: a1 = p1 by A834,A835,A885,A888,GOBOARD1:5; right_cell(f,m-'1,G) = cell(G,a1,a2) by A428,A572,A843,A850,A851 ,A883,A884,A885,A888,GOBRD13:22 .= left_cell(f,l,G) by A428,A569,A580,A843,A846,A848,A845,A834 ,A835,A836,A837,A882,A890,A889,GOBRD13:25; hence thesis by A428,A580,A846,A848,A852; end; suppose A891: a1+1 = p91 & a2 = p92; A892: m-'1 <= m by A850,NAT_1:11; A893: f/.(m-'1) = (F.(m-'1))/.(m-'1) by A580,A577,A792,A851 .= (F.m)/.(m-'1) by A792,A851,A892; A894: 2 in dom g by A623,FINSEQ_3:25; len rg-'1+2 = len g +1 by A824,FINSEQ_5:def 3; then A895: rg/.(len rg-'1) = g/.2 by A894,FINSEQ_5:66 .= f/.(m+1) by A826,A894,FINSEQ_5:27; A896: p = g/.1 by A569,A614,A579,FINSEQ_5:65 .= rg/.len g by FINSEQ_5:65 .= rg/.len rg by FINSEQ_5:def 3; A897: a2 = p2 by A834,A835,A885,A891,GOBOARD1:5; A898: m-'1+1 <= len (F.m) & f/.(m-'1+1) = (F.m)/.m by A192,A580,A570 ,A572,A792,A850; A899: F.k|(m+1)=F.(m+1) by A490,A580,A576; set rc = right_cell(rg,len rg-'1,G)\L~rg; A900: a1 < a1+1 & p1 < p1+1 by XREAL_1:29; A901: a1+1 = p1 by A834,A835,A885,A891,GOBOARD1:5; then A902: a1 = p1-'1 by NAT_D:34; right_cell(f,l,G) = cell(G,p1,p2) by A428,A569,A580,A843,A846 ,A848,A845,A834,A835,A836,A837,A882,GOBRD13:26 .= front_left_cell(F.m,m-'1,G) by A428,A843,A850,A851,A834,A835 ,A883,A884,A901,A897,A893,A898,GOBRD13:36; then F.(m+1) turns_left m-'1,G by A458,A796,A849; then A903: f turns_left m-'1,G by A851,A826,A899,GOBRD13:44; then A904: [p1,p2+1] in Indices G by A843,A850,A834,A835,A883,A884,A901,A900 ,GOBRD13:def 7; then A905: p2+1 <= width G by MATRIX_1:38; f/.(m+1) = G*(p1,p2+1) by A843,A850,A826,A834,A835,A883,A884,A901 ,A900,A903,GOBRD13:def 7; then right_cell(rg,len rg-'1,G) = cell(G,p1-'1,a2) by A833,A831 ,A832,A834,A835,A897,A904,A895,A896,GOBRD13:28; then a in right_cell(rg,len rg-'1,G) by A841,A842,A884,A887,A901 ,A897,A905,A902,Th20; then A906: a in rc by A854,XBOOLE_0:def 5; A907: RightComp rg is_a_component_of (L~rg)` by GOBOARD9:def 2; rc c= RightComp rg & L~rg c= L~f by A791,A833,A831,A832,Th27, SPPOL_2:22; hence thesis by A572,A773,A850,A851,A854,A906,A907; end; suppose a1 = p91+1 & a2 = p92; then A908: a1 = q1 & a2 = q2 by A834,A835,A882,A885,GOBOARD1:5; (F.l)/.(m-'1) = (F.(m-'1))/.(m-'1) by A792,A851,NAT_1:11 .= q by A580,A577,A792,A851,A837,A884,A908 .= (F.l)/.l by A792,A847,A848,A845 .= (F.l)/.len(F.l) by A192; hence thesis by A566,A847,A848,A853,A829; end; suppose a1 = p91 & a2 = p92+1; then A909: a1 = p1 & a2 = p2+1 by A834,A835,A885,GOBOARD1:5; right_cell(f,l,G) = cell(G,p1,p2) by A428,A569,A580,A843,A846 ,A848,A845,A834,A835,A836,A837,A882,GOBRD13:26 .= left_cell(f,m-'1,G) by A428,A572,A843,A850,A851,A834,A835 ,A883,A884,A909,GOBRD13:27; hence thesis by A428,A572,A850,A851,A849; end; end; end; suppose A910: p1 = q1+1 & p2 = q2; consider a1,a2,p91,p92 being Element of NAT such that A911: [a1,a2] in Indices G and A912: a = G*(a1,a2) and A913: [p91,p92] in Indices G & p = G*(p91,p92) and A914: a1 = p91 & a2+1 = p92 or a1+1 = p91 & a2 = p92 or a1 = p91+1 & a2 = p92 or a1 = p91 & a2 = p92+1 by A571,A572,A843,A850,A851,JORDAN8:3 ; A915: a1 <= len G by A911,MATRIX_1:38; thus thesis proof per cases by A914; suppose A916: a1 = p91 & a2+1 = p92; then a2+1 = p2 by A834,A835,A913,GOBOARD1:5; then A917: q2-'1 = a2 by A910,NAT_D:34; a1 = p1 by A834,A835,A913,A916,GOBOARD1:5; then A918: q1 = a1-'1 by A910,NAT_D:34; right_cell(f,l,G) = cell(G,q1,q2-'1) by A428,A569,A580,A843,A846 ,A848,A845,A834,A835,A836,A837,A910,GOBRD13:24 .= left_cell(f,m-'1,G) by A428,A572,A843,A850,A851,A911,A912 ,A913,A916,A918,A917,GOBRD13:21; hence thesis by A428,A572,A850,A851,A849; end; suppose a1+1 = p91 & a2 = p92; then A919: a1+1 = p1 & a2 = p2 by A834,A835,A913,GOBOARD1:5; (F.l)/.(m-'1) = (F.(m-'1))/.(m-'1) by A792,A851,NAT_1:11 .= q by A580,A577,A792,A851,A837,A910,A912,A919 .= (F.l)/.l by A792,A847,A848,A845 .= (F.l)/.len(F.l) by A192; hence thesis by A566,A847,A848,A853,A829; end; suppose A920: a1 = p91+1 & a2 = p92; A921: p = g/.1 by A569,A614,A579,FINSEQ_5:65 .= rg/.len g by FINSEQ_5:65 .= rg/.len rg by FINSEQ_5:def 3; A922: a1 = p1+1 by A834,A835,A913,A920,GOBOARD1:5; p1+1>p1 by XREAL_1:29; then A923: a1+1 > p1 by A922,NAT_1:13; A924: m-'1 <= m by A850,NAT_1:11; A925: f/.(m-'1) = (F.(m-'1))/.(m-'1) by A580,A577,A792,A851 .= (F.m)/.(m-'1) by A792,A851,A924; A926: 2 in dom g by A623,FINSEQ_3:25; len rg-'1+2 = len g +1 by A824,FINSEQ_5:def 3; then A927: rg/.(len rg-'1) = g/.2 by A926,FINSEQ_5:66 .= f/.(m+1) by A826,A926,FINSEQ_5:27; A928: F.k|(m+1)=F.(m+1) by A490,A580,A576; set rc = right_cell(rg,len rg-'1,G)\L~rg; A929: RightComp rg is_a_component_of (L~rg)` by GOBOARD9:def 2; A930: p2-'1+1 = p2 by A842,XREAL_1:235; A931: f/.(m-'1+1) = (F.m)/.m by A580,A570,A572,A792,A850; A932: p1-'1 = q1 & m-'1+1 <= len (F.m) by A192,A850,A910,NAT_D:34; A933: a2 = p2 by A834,A835,A913,A920,GOBOARD1:5; right_cell(f,l,G) = cell(G,q1,q2-'1) by A428,A569,A580,A843,A846 ,A848,A845,A834,A835,A836,A837,A910,GOBRD13:24 .= front_left_cell(F.m,m-'1,G) by A428,A843,A850,A851,A834,A835 ,A910,A911,A912,A922,A933,A932,A925,A931,GOBRD13:38; then F.(m+1) turns_left m-'1,G by A458,A796,A849; then A934: f turns_left m-'1,G by A851,A826,A928,GOBRD13:44; A935: a2+1 > p2 by A933,NAT_1:13; then A936: [p1,p2-'1] in Indices G by A843,A850,A834,A835,A911,A912,A933 ,A923,A934,GOBRD13:def 7; then A937: 1 <= p2-'1 by MATRIX_1:38; f/.(m+1) = G*(p1,p2-'1) by A843,A850,A826,A834,A835,A911,A912 ,A933,A935,A923,A934,GOBRD13:def 7; then right_cell(rg,len rg-'1,G) = cell(G,p1,p2-'1) by A833,A831 ,A832,A834,A835,A936,A930,A927,A921,GOBRD13:22; then a in right_cell(rg,len rg-'1,G) by A839,A840,A912,A915,A922 ,A933,A937,A930,Th20; then A938: a in rc by A854,XBOOLE_0:def 5; rc c= RightComp rg & L~rg c= L~f by A791,A833,A831,A832,Th27, SPPOL_2:22; hence thesis by A572,A773,A850,A851,A854,A938,A929; end; suppose A939: a1 = p91 & a2 = p92+1; then a1 = p1 by A834,A835,A913,GOBOARD1:5; then A940: q1 = a1-'1 by A910,NAT_D:34; a2 = p2+1 by A834,A835,A913,A939,GOBOARD1:5; then right_cell(f,m-'1,G) = cell(G,q1,q2) by A428,A572,A843,A850 ,A851,A910,A911,A912,A913,A939,A940,GOBRD13:28 .= left_cell(f,l,G) by A428,A569,A580,A843,A846,A848,A845,A834 ,A835,A836,A837,A910,GOBRD13:23; hence thesis by A428,A580,A846,A848,A852; end; end; end; suppose A941: p1 = q1 & p2 = q2+1; consider a1,a2,p91,p92 being Element of NAT such that A942: [a1,a2] in Indices G and A943: a = G*(a1,a2) and A944: [p91,p92] in Indices G & p = G*(p91,p92) and A945: a1 = p91 & a2+1 = p92 or a1+1 = p91 & a2 = p92 or a1 = p91+1 & a2 = p92 or a1 = p91 & a2 = p92+1 by A571,A572,A843,A850,A851,JORDAN8:3 ; A946: a2 <= width G by A942,MATRIX_1:38; thus thesis proof per cases by A945; suppose a1 = p91 & a2+1 = p92; then A947: a1 = p1 & a2+1 = p2 by A834,A835,A944,GOBOARD1:5; (F.l)/.(m-'1) = (F.(m-'1))/.(m-'1) by A792,A851,NAT_1:11 .= q by A580,A577,A792,A851,A837,A941,A943,A947 .= (F.l)/.l by A792,A847,A848,A845 .= (F.l)/.len(F.l) by A192; hence thesis by A566,A847,A848,A853,A829; end; suppose A948: a1+1 = p91 & a2 = p92; then a2 = p2 by A834,A835,A944,GOBOARD1:5; then A949: a2-'1 = q2 by A941,NAT_D:34; a1+1 = p1 by A834,A835,A944,A948,GOBOARD1:5; then A950: a1 = q1-'1 by A941,NAT_D:34; right_cell(f,m-'1,G) = cell(G,a1,a2-'1) by A428,A572,A843,A850 ,A851,A942,A943,A944,A948,GOBRD13:24 .= left_cell(f,l,G) by A428,A569,A580,A843,A846,A848,A845,A834 ,A835,A836,A837,A941,A950,A949,GOBRD13:21; hence thesis by A428,A580,A846,A848,A852; end; suppose A951: a1 = p91+1 & a2 = p92; then a2 = p2 by A834,A835,A944,GOBOARD1:5; then A952: a2-'1 = q2 by A941,NAT_D:34; A953: a1 = p1+1 by A834,A835,A944,A951,GOBOARD1:5; right_cell(f,l,G) = cell(G,q1,q2) by A428,A569,A580,A843,A846 ,A848,A845,A834,A835,A836,A837,A941,GOBRD13:22 .= left_cell(f,m-'1,G) by A428,A572,A843,A850,A851,A941,A942 ,A943,A944,A951,A953,A952,GOBRD13:25; hence thesis by A428,A572,A850,A851,A849; end; suppose A954: a1 = p91 & a2 = p92+1; set rc = right_cell(rg,len rg-'1,G)\L~rg; A955: RightComp rg is_a_component_of (L~rg)` by GOBOARD9:def 2; A956: 2 in dom g by A623,FINSEQ_3:25; len rg-'1+2 = len g +1 by A824,FINSEQ_5:def 3; then A957: rg/.(len rg-'1) = g/.2 by A956,FINSEQ_5:66 .= f/.(m+1) by A826,A956,FINSEQ_5:27; A958: p = g/.1 by A569,A614,A579,FINSEQ_5:65 .= rg/.len g by FINSEQ_5:65 .= rg/.len rg by FINSEQ_5:def 3; A959: a1 = p1 by A834,A835,A944,A954,GOBOARD1:5; A960: m-'1 <= m by A850,NAT_1:11; A961: f/.(m-'1) = (F.(m-'1))/.(m-'1) by A580,A577,A792,A851 .= (F.m)/.(m-'1) by A792,A851,A960; A962: p2-'1 = q2 & m-'1+1 <= len (F.m) by A192,A850,A941,NAT_D:34; A963: f/.(m-'1+1) = (F.m)/.m by A580,A570,A572,A792,A850; A964: F.k|(m+1)=F.(m+1) by A490,A580,A576; A965: a2 = p2+1 by A834,A835,A944,A954,GOBOARD1:5; right_cell(f,l,G) = cell(G,q1,q2) by A428,A569,A580,A843,A846 ,A848,A845,A834,A835,A836,A837,A941,GOBRD13:22 .= front_left_cell(F.m,m-'1,G) by A428,A843,A850,A851,A834,A835 ,A941,A942,A943,A959,A965,A962,A961,A963,GOBRD13:40; then F.(m+1) turns_left m-'1,G by A458,A796,A849; then A966: p2+1>p2 & f turns_left m-'1,G by A851,A826,A964,GOBRD13:44 ,NAT_1:13; A967: a2+1>p2+1 by A965,NAT_1:13; then A968: [p1+1,p2] in Indices G by A843,A850,A834,A835,A942,A943,A966, GOBRD13:def 7; then A969: p1+1 <= len G by MATRIX_1:38; f/.(m+1) = G*(p1+1,p2) by A843,A850,A826,A834,A835,A942,A943,A967 ,A966,GOBRD13:def 7; then right_cell(rg,len rg-'1,G) = cell(G,p1,p2) by A833,A831,A832 ,A834,A835,A968,A957,A958,GOBRD13:26; then a in right_cell(rg,len rg-'1,G) by A839,A842,A943,A946,A959 ,A965,A969,Th20; then A970: a in rc by A854,XBOOLE_0:def 5; rc c= RightComp rg & L~rg c= L~f by A791,A833,A831,A832,Th27, SPPOL_2:22; hence thesis by A572,A773,A850,A851,A854,A970,A955; end; end; end; end; then A971: LeftComp g is_a_component_of (L~g)` & C meets LeftComp g by GOBOARD9:24 ,def 1; reconsider Lg9 = (L~g)` as Subset of TOP-REAL 2; A972: RightComp g is_a_component_of (L~g)` by GOBOARD9:def 2; A973: C c= Lg9 proof let c be set; assume that A974: c in C and A975: not c in Lg9; reconsider c as Point of TOP-REAL 2 by A974; consider i such that A976: 1 <= i and A977: i+1 <= len g and A978: c in LSeg(g/.i,g/.(i+1)) by A975,SPPOL_2:14,SUBSET_1:29; A979: 1 <= i+(m-'1) by A976,NAT_1:12; i+1 in dom g by A976,A977,SEQ_4:134; then A980: g/.(i+1) = f/.(i+1+(m-'1)) by FINSEQ_5:27; i+1+(m-'1) = i+(m-'1)+1; then A981: i+(m-'1)+1 <= (len g)+(m-'1) by A977,XREAL_1:6; i in dom g by A976,A977,SEQ_4:134; then g/.i = f/.(i+(m-'1)) by FINSEQ_5:27; then c in LSeg(f,i+(m-'1)) by A578,A978,A980,A979,A981,TOPREAL1:def 3; then c in left_cell(f,i+(m-'1),G) /\ right_cell(f,i+(m-'1),G) by A428 ,A578,A979,A981,GOBRD13:29; then c in left_cell(f,i+(m-'1),G) by XBOOLE_0:def 4; then left_cell(f,i+(m-'1),G) meets C by A974,XBOOLE_0:3; hence contradiction by A428,A578,A979,A981; end; A982: the TopStruct of TOP-REAL 2 = TopSpaceMetr (Euclid 2) by EUCLID:def 8; C meets RightComp g proof right_cell(f,m,G) meets C by A428,A570,A576; then consider p being set such that A983: p in right_cell(f,m,G) and A984: p in C by XBOOLE_0:3; reconsider p as Element of TOP-REAL 2 by A983; now reconsider u = p as Element of Euclid 2 by TOPREAL3:8; take a = p; thus a in C by A984; consider r being real number such that A985: r > 0 and A986: Ball(u,r) c= (L~g)` by A973,A790,A984,GOBOARD6:5; reconsider r as Real by XREAL_0:def 1; A987: p in Ball(u,r) by A985,GOBOARD6:1; reconsider B = Ball(u,r) as non empty Subset of TOP-REAL 2 by A982 ,A985,GOBOARD6:1,TOPMETR:12; A988: p in B by A985,GOBOARD6:1; right_cell(f,m,G) = Cl Int right_cell(f,m,G) & B is open by A570,A576 ,A756,GOBOARD6:3; then A989: Int right_cell(f,m,G) meets B by A983,A987,TOPS_1:12; Int right_cell(g,1,G) c= Int right_cell(g,1) & Int right_cell (g,1) c= RightComp g by A625,A624,GOBOARD9:25,GOBRD13:33,TOPS_1:19; then Int right_cell(g,1,G) c= RightComp g by XBOOLE_1:1; then Int right_cell(f,m-'1+1,G) c= RightComp g by A571,A577,A624, GOBRD13:32; then B is connected & Int right_cell(f,m,G) c= RightComp g by A570, SPRECT_3:7,XREAL_1:235; then B c= RightComp g by A972,A986,A989,GOBOARD9:4; hence a in RightComp g by A988; end; hence thesis by XBOOLE_0:3; end; hence contradiction by A1,A972,A973,A971,Th1,SPRECT_4:6; end; then g = f/^0 by XREAL_1:232 .= f by FINSEQ_5:28; then reconsider f as standard non constant special_circular_sequence; f is clockwise_oriented proof f/.2 in LSeg(f/.1,f/.(1+1)) by RLTOPSP1:68; then A990: f/.2 in L~f by A574,SPPOL_2:15; (NW-corner L~f)`1 = W-bound L~f by EUCLID:52; then A991: (NW-corner L~f)`1 <= (f/.2)`1 by A990,PSCOMP_1:24; len G >= 3 by A231,NAT_1:12; then A992: 1 < len G by XXREAL_0:2; (NE-corner L~f)`1 = E-bound L~f by EUCLID:52; then A993: (f/.2)`1 <= (NE-corner L~f)`1 by A990,PSCOMP_1:24; for k st 1 <= k & k+1 <= len f holds left_cell(f,k,G) misses C & right_cell(f,k,G) meets C by A428; then A994: N-min L~f = f/.1 by A571,A608,Th30; consider i such that A995: 1 <= i and A996: i+1 <= len G and A997: f/.1 = G*(i,width G) & f/.2 = G*(i+1,width G) and N-min C in cell(G,i,width G-'1) and N-min C <> G*(i,width G-'1) by A608; i < len G by A996,NAT_1:13; then A998: (N-min L~f)`2 = N-bound L~f & G*(i,width G)`2 = G*(1,width G)`2 by A2,A992,A995,EUCLID:52,GOBOARD5:1; 1 <= i+1 by NAT_1:12; then A999: G*(i+1,width G)`2 = G*(1,width G)`2 by A2,A992,A996,GOBOARD5:1; (NW-corner L~f)`2 = (NE-corner L~f)`2 & (NE-corner L~f)`2 = N-bound L~f by EUCLID:52,PSCOMP_1:27; then f/.2 in LSeg(NW-corner L~f, NE-corner L~f) by A994,A997,A998,A999 ,A991,A993,GOBOARD7:8; then f/.2 in LSeg(NW-corner L~f, NE-corner L~f) /\ L~f by A990, XBOOLE_0:def 4; hence thesis by A994,SPRECT_2:30; end; then reconsider f as clockwise_oriented standard non constant special_circular_sequence; take f; thus f is_sequence_on G by A428; thus ex i st 1 <= i & i+1 <= len G & f/.1 = G*(i,width G) & f/.2 = G*(i+1, width G) & N-min C in cell(G,i,width G-'1) & N-min C <> G*(i,width G-'1) by A608; let m such that A1000: 1 <= m and A1001: m+2 <= len f; A1002: F.(m+1+1) = f|(m+1+1) by A490,A580,A1001; A1003: m+1 < m+2 by XREAL_1:6; then A1004: f|(m+1) = F.(m+1) by A490,A580,A1001,XXREAL_0:2; A1005: m+1 <= len f by A1001,A1003,XXREAL_0:2; then A1006: front_left_cell(F.(m+1),m,G) = front_left_cell(f,m,G) by A571,A1000 ,A1004,GOBRD13:42; A1007: m = m+1-'1 & m+1 > 1 by A1000,NAT_1:13,NAT_D:34; A1008: front_right_cell(F.(m+1),m,G) = front_right_cell(f,m,G) by A571,A1000 ,A1005,A1004,GOBRD13:42; hereby assume front_left_cell(f,m,G) misses C & front_right_cell(f,m,G) misses C; then F.(m+1+1) turns_right m,G by A458,A1007,A1006,A1008; hence f turns_right m,G by A1000,A1001,A1002,GOBRD13:43; end; hereby assume front_left_cell(f,m,G) misses C & front_right_cell(f,m,G) meets C; then F.(m+1+1) goes_straight m,G by A458,A1007,A1006,A1008; hence f goes_straight m,G by A1000,A1001,A1002,GOBRD13:45; end; assume front_left_cell(f,m,G) meets C; then F.(m+1+1) turns_left m,G by A458,A1007,A1006; hence thesis by A1000,A1001,A1002,GOBRD13:44; end; uniqueness proof let f1,f2 be clockwise_oriented standard non constant special_circular_sequence such that A1009: f1 is_sequence_on Gauge(C,n); defpred P[Element of NAT] means f1|$1 = f2|$1; given i1 such that A1010: 1 <= i1 & i1+1 <= len Gauge(C,n) and A1011: f1/.1 = Gauge(C,n)*(i1,width Gauge(C,n)) and A1012: f1/.2 = Gauge(C,n)*(i1+1,width Gauge(C,n)) and A1013: N-min C in cell(Gauge(C,n),i1,width Gauge(C,n)-'1) & N-min C <> Gauge(C,n)*( i1,width Gauge(C,n)-'1); assume that A1014: for k st 1 <= k & k+2 <= len f1 holds (front_left_cell(f1,k, Gauge(C,n)) misses C & front_right_cell(f1,k,Gauge(C,n)) misses C implies f1 turns_right k,Gauge(C,n)) & (front_left_cell(f1,k,Gauge(C,n)) misses C & front_right_cell(f1,k,Gauge(C,n)) meets C implies f1 goes_straight k,Gauge(C,n) ) & (front_left_cell(f1,k,Gauge(C,n)) meets C implies f1 turns_left k,Gauge(C,n )) and A1015: f2 is_sequence_on Gauge(C,n); given i2 such that A1016: 1 <= i2 & i2+1 <= len Gauge(C,n) and A1017: f2/.1 = Gauge(C,n)*(i2,width Gauge(C,n)) and A1018: f2/.2 = Gauge(C,n)*(i2+1,width Gauge(C,n)) and A1019: N-min C in cell(Gauge(C,n),i2,width Gauge(C,n)-'1) & N-min C <> Gauge(C,n)*( i2,width Gauge(C,n)-'1); assume A1020: for k st 1 <= k & k+2 <= len f2 holds (front_left_cell(f2,k, Gauge(C,n)) misses C & front_right_cell(f2,k,Gauge(C,n)) misses C implies f2 turns_right k,Gauge(C,n)) & (front_left_cell(f2,k,Gauge(C,n)) misses C & front_right_cell(f2,k,Gauge(C,n)) meets C implies f2 goes_straight k,Gauge(C,n) ) & (front_left_cell(f2,k,Gauge(C,n)) meets C implies f2 turns_left k,Gauge(C,n )); A1021: for k st P[k] holds P[k+1] proof A1022: len f1 > 4 by GOBOARD7:34; A1023: f1|1 = <*f1/.1*> & f2|1 = <*f2/.1*> by FINSEQ_5:20; A1024: i1 = i2 & len f2 > 4 by A1010,A1013,A1016,A1019,Th29,GOBOARD7:34; let k such that A1025: f1|k = f2|k; per cases by NAT_1:25; suppose k = 0; hence thesis by A1010,A1011,A1013,A1016,A1017,A1019,A1023,Th29; end; suppose A1026: k = 1; f1|2 = <*f1/.1,f1/.2*> by A1022,FINSEQ_5:81,XXREAL_0:2; hence thesis by A1011,A1012,A1017,A1018,A1024,A1026,FINSEQ_5:81 ,XXREAL_0:2; end; suppose A1027: k > 1; A1028: f1/.1 = f1/.len f1 & f2/.1 = f2/.len f2 by FINSEQ_6:def 1; now per cases; suppose A1029: len f1 > k; set m = k-'1; A1030: 1 <= m by A1027,NAT_D:49; then A1031: m+1 = k by NAT_D:43; then A1032: front_left_cell (f1,m,Gauge(C,n))= front_left_cell(f1|k,m, Gauge(C,n)) by A1009,A1029,A1030,GOBRD13:42; A1033: m+(1+1) = k+1 by A1031; A1034: k+1 <= len f1 by A1029,NAT_1:13; A1035: now A1036: 1 < len f2 by GOBOARD7:34,XXREAL_0:2; assume A1037: len f2 <= k; then A1038: f2 = f2|k by FINSEQ_1:58; then len f2 in dom(f2|k) by FINSEQ_5:6; then A1039: (f1|k)/.len f2 = f1/.len f2 by A1025,FINSEQ_4:70; 1 in dom(f2|k) & len f2 <= len f1 by A1025,A1038,FINSEQ_5:6,16; hence contradiction by A1025,A1028,A1029,A1037,A1038,A1039,A1036, FINSEQ_4:70,GOBOARD7:38; end; then A1040: k+1 <= len f2 by NAT_1:13; A1041: front_right_cell(f2,m,Gauge(C,n)) = front_right_cell(f2|k,m ,Gauge(C,n)) by A1015,A1030,A1031,A1035,GOBRD13:42; A1042: front_left_cell(f2,m,Gauge(C,n)) = front_left_cell(f2|k,m, Gauge(C,n)) by A1015,A1030,A1031,A1035,GOBRD13:42; A1043: front_right_cell(f1,m,Gauge(C,n)) = front_right_cell(f1|k,m ,Gauge(C,n)) by A1009,A1029,A1030,A1031,GOBRD13:42; now per cases; suppose front_left_cell(f1,m,Gauge(C,n)) misses C & front_right_cell(f1,m,Gauge(C,n)) misses C; then f1 turns_right m,Gauge(C,n) & f2 turns_right m,Gauge(C, n) by A1014,A1020,A1025,A1030,A1040,A1034,A1032,A1043,A1042,A1041,A1033; hence thesis by A1015,A1025,A1027,A1040,A1034,GOBRD13:46; end; suppose front_left_cell(f1,m,Gauge(C,n)) misses C & front_right_cell(f1,m,Gauge(C,n)) meets C; then f1 goes_straight m,Gauge(C,n) & f2 goes_straight m, Gauge(C,n) by A1014,A1020,A1025,A1030,A1040,A1034,A1032,A1043,A1042,A1041,A1033 ; hence thesis by A1015,A1025,A1027,A1040,A1034,GOBRD13:48; end; suppose front_left_cell(f1,m,Gauge(C,n)) meets C; then f1 turns_left m,Gauge(C,n) & f2 turns_left m,Gauge(C,n) by A1014,A1020,A1025,A1030,A1040,A1034,A1032,A1042,A1033; hence thesis by A1015,A1025,A1027,A1040,A1034,GOBRD13:47; end; end; hence thesis; end; suppose A1044: k >= len f1; A1045: 1 < len f1 by GOBOARD7:34,XXREAL_0:2; A1046: f1 = f1|k by A1044,FINSEQ_1:58; then len f1 in dom(f1|k) by FINSEQ_5:6; then A1047: (f2|k)/.len f1 = f2/.len f1 by A1025,FINSEQ_4:70; 1 in dom(f1|k) & len f1 <= len f2 by A1025,A1046,FINSEQ_5:6,16; then A1048: len f2 = len f1 by A1025,A1028,A1046,A1047,A1045,FINSEQ_4:70 ,GOBOARD7:38; A1049: k+1 > len f1 by A1044,NAT_1:13; hence f1|(k+1) = f1 by FINSEQ_1:58 .= f2 by A1025,A1044,A1046,A1048,FINSEQ_1:58 .= f2|(k+1) by A1048,A1049,FINSEQ_1:58; end; end; hence thesis; end; end; A1050: P[0]; for k holds P[k] from NAT_1:sch 1(A1050,A1021); hence thesis by Th2; end; end; theorem Th31: C is connected & 1 <= k & k+1 <= len Cage(C,n) implies left_cell (Cage(C,n),k,Gauge(C,n)) misses C & right_cell(Cage(C,n),k,Gauge(C,n)) meets C proof set G = Gauge(C,n), f = Cage(C,n); set W = W-bound C, E = E-bound C, S = S-bound C, N = N-bound C; defpred P[Element of NAT] means for m st 1 <= m & m+1 <= len(f|$1) holds left_cell(f|$1,m,G) misses C & right_cell(f|$1,m,G) meets C; A1: len G = width G by JORDAN8:def 1; assume A2: C is connected; then A3: f is_sequence_on G by Def1; A4: len G = 2|^n+3 by JORDAN8:def 1; A5: for k st P[k] holds P[k+1] proof let k such that A6: for m st 1 <= m & m+1 <= len(f|k) holds left_cell(f|k,m,G) misses C & right_cell(f|k,m,G) meets C; per cases; suppose k >= len f; then f|k = f & f|(k+1) = f by FINSEQ_1:58,NAT_1:12; hence thesis by A6; end; suppose A7: k < len f; then A8: len(f|k) = k by FINSEQ_1:59; A9: 1 <= len G by A4,NAT_1:12; A10: f|k is_sequence_on G by A3,GOBOARD1:22; A11: f|(k+1) is_sequence_on G by A3,GOBOARD1:22; consider i such that A12: 1 <= i and A13: i+1 <= len G and A14: f/.1 = G*(i,width G) & f/.2 = G*(i+1,width G) and A15: N-min C in cell(G,i,width G-'1) and N-min C <> G*(i,width G-'1) by A2,Def1; let m such that A16: 1 <= m and A17: m+1 <= len(f|(k+1)); A18: k+1 <= len f by A7,NAT_1:13; then A19: len(f|(k+1)) = k+1 by FINSEQ_1:59; A20: 2|^n >= n+1 by NEWTON:85; now per cases by NAT_1:25; suppose A21: k = 0; 1 <= m+1 by NAT_1:12; then m+1 = 0+1 by A19,A17,A21,XXREAL_0:1; hence thesis by A16; end; suppose A22: k = 1; 1+1 <= m+1 by A16,XREAL_1:6; then A23: m+1 = 1+1 by A19,A17,A22,XXREAL_0:1; f|(k+1) = <*G*(i,width G),G*(i+1,width G)*> by A18,A14,A22, FINSEQ_5:81; then A24: (f|(k+1))/.1 = G*(i,width G) & (f|(k+1))/.2 = G*(i+1, width G) by FINSEQ_4:17; 1 <= i+1 by A12,NAT_1:13; then A25: [i+1,len G] in Indices G by A1,A13,A9,MATRIX_1:36; A26: i < len G by A13,NAT_1:13; then A27: [i,len G] in Indices G by A1,A12,A9,MATRIX_1:36; A28: i < i+1 & i+1 < (i+1)+1 by NAT_1:13; then A29: left_cell(f|(k+1),m,G) = cell(G,i,len G) by A1,A11,A17,A24,A27,A25 ,A23,GOBRD13:def 3; now N > S by JORDAN8:9; then N-S > S-S by XREAL_1:9; then (N-S)/(2|^n) > 0 by A20,XREAL_1:139; then A30: N+0 < N+(N-S)/(2|^n) by XREAL_1:6; assume left_cell(f|(k+1),m,G) meets C; then consider p being set such that A31: p in cell(G,i,len G) and A32: p in C by A29,XBOOLE_0:3; reconsider p as Element of TOP-REAL 2 by A31; A33: p`2 <= N by A32,PSCOMP_1:24; [1,len G] in Indices G by A1,A9,MATRIX_1:36; then G*(1,len G) = |[W+((E-W)/(2|^n))*(1-2),S+((N-S)/(2 |^n))*(( len G)- 2)]| by JORDAN8:def 1; then A34: G*(1,len G)`2 = S+((N-S)/(2|^n))*((len G)-2) by EUCLID:52; cell(G,i,len G) = { |[r,s]|: G*(i,1)`1 <= r & r <= G*(i+1,1) `1 & G*(1,len G)`2 <= s } by A1,A12,A26,GOBRD11:31; then consider r,s such that A35: p = |[r,s]| and G*(i,1)`1 <= r and r <= G*(i+1,1)`1 and A36: G*(1,len G)`2 <= s by A31; ((N-S)/(2|^n))*((len G)-2) = ((N-S)/(2|^n))*(2|^n)+((N-S)/(2 |^n))*1 by A4 .= (N-S)+(N-S)/(2|^n) by A20,XCMPLX_1:87; then N < s by A36,A34,A30,XXREAL_0:2; hence contradiction by A35,A33,EUCLID:52; end; hence left_cell(f|(k+1),m,G) misses C; N-min C in C & N-min C in right_cell(f|(k+1),m,G) by A1,A11,A15,A17 ,A24,A27,A25,A23,A28,GOBRD13:def 2,SPRECT_1:11; hence right_cell(f|(k+1),m,G) meets C by XBOOLE_0:3; end; suppose A37: k > 1; then A38: (len(f|k)) -'1 +1 = len(f|k) by A8,XREAL_1:235; A39: 1 <= (len(f|k))-'1 by A8,A37,NAT_D:49; now per cases; suppose A40: m+1 = len(f|(k+1)); A41: len(f|k) <= len f by FINSEQ_5:16; now left_cell(f|k,(len(f|k))-'1,G) misses C by A6,A39,A38; then A42: left_cell(f,(len(f|k))-'1,G) misses C by A3,A8,A39,A38,A41, GOBRD13:31; A43: (len(f|k))-'1+(1+1) = (len(f|k))+1 by A38; right_cell(f|k,(len(f|k))-'1,G) meets C by A6,A39,A38; then A44: right_cell(f,(len(f|k))-'1,G) meets C by A3,A8,A39,A38,A41, GOBRD13:31; consider i1,j1,i2,j2 being Element of NAT such that A45: [i1,j1] in Indices G & f/.((len(f|k)) -'1) = G*(i1,j1 ) and A46: [i2,j2] in Indices G and A47: f/.len(f|k) = G*(i2,j2) and i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A3,A7,A8,A39,A38,JORDAN8:3; 1 <= i2 by A46,MATRIX_1:38; then A48: (i2-'1)+1 = i2 by XREAL_1:235; 1 <= j2 by A46,MATRIX_1:38; then A49: (j2-'1)+1 = j2 by XREAL_1:235; per cases; suppose A50: front_left_cell(f,(len(f|k))-'1,G) misses C & front_right_cell(f,(len(f|k))-'1,G) misses C; then A51: f turns_right (len(f|k))-'1,G by A2,A18,A8,A39,A43,Def1; now per cases by A38,A45,A46,A47,A43,A51,GOBRD13:def 6; suppose that A52: i1 = i2 & j1+1 = j2 and A53: [i2+1,j2] in Indices G & f/.((len(f|k))+1) = G* (i2+1,j2); front_right_cell(f,(len(f|k))-'1,G) = cell(G,i1,j2) by A3,A39,A38,A41,A45,A46,A47,A52,GOBRD13:35; then left_cell(f,m,G) misses C by A3,A18,A8,A19,A16,A40,A46 ,A47,A50,A52,A53,GOBRD13:23; hence left_cell(f|(k+1),m,G) misses C by A3,A18,A19,A16 ,A40,GOBRD13:31; j2-'1 = j1 & cell(G,i1,j1) meets C by A3,A39,A38,A41,A45 ,A46,A47,A44,A52,GOBRD13:22,NAT_D:34; then right_cell(f,m,G) meets C by A3,A18,A8,A19,A16,A40,A46 ,A47,A52,A53,GOBRD13:24; hence right_cell(f|(k+1),m,G) meets C by A3,A18,A19,A16 ,A40,GOBRD13:31; end; suppose that A54: i1+1 = i2 & j1 = j2 and A55: [i2,j2-'1] in Indices G & f/.((len(f|k))+1) = G *(i2,j2-'1); front_right_cell(f,(len(f|k))-'1,G) = cell(G,i2,j2 -'1) by A3,A39,A38,A41,A45,A46,A47,A54,GOBRD13:37; then left_cell(f,m,G) misses C by A3,A18,A8,A19,A16,A40,A46 ,A47,A49,A50,A55,GOBRD13:27; hence left_cell(f|(k+1),m,G) misses C by A3,A18,A19,A16 ,A40,GOBRD13:31; i2-'1 = i1 & cell(G,i1,j1-'1) meets C by A3,A39,A38,A41 ,A45,A46,A47,A44,A54,GOBRD13:24,NAT_D:34; then right_cell(f,m,G) meets C by A3,A18,A8,A19,A16,A40,A46 ,A47,A49,A54,A55,GOBRD13:28; hence right_cell(f|(k+1),m,G) meets C by A3,A18,A19,A16 ,A40,GOBRD13:31; end; suppose that A56: i1 = i2+1 & j1 = j2 and A57: [i2,j2+1] in Indices G & f/.((len(f|k))+1) = G* (i2,j2+1); front_right_cell(f,(len(f|k))-'1,G) = cell(G,i2-'1, j2) by A3,A39,A38,A41,A45,A46,A47,A56,GOBRD13:39; then left_cell(f,m,G) misses C by A3,A18,A8,A19,A16,A40,A46 ,A47,A50,A57,GOBRD13:21; hence left_cell(f|(k+1),m,G) misses C by A3,A18,A19,A16 ,A40,GOBRD13:31; cell(G,i2,j2) meets C by A3,A39,A38,A41,A45,A46,A47,A44 ,A56,GOBRD13:26; then right_cell(f,m,G) meets C by A3,A18,A8,A19,A16,A40,A46 ,A47,A57,GOBRD13:22; hence right_cell(f|(k+1),m,G) meets C by A3,A18,A19,A16 ,A40,GOBRD13:31; end; suppose that A58: i1 = i2 & j1 = j2+1 and A59: [i2-'1,j2] in Indices G & f/.((len(f|k))+1) = G *(i2-'1,j2); front_right_cell(f,(len(f|k))-'1,G) = cell(G,i2-'1, j2-'1 ) by A3,A39,A38,A41,A45,A46,A47,A58,GOBRD13:41; then left_cell(f,m,G) misses C by A3,A18,A8,A19,A16,A40 ,A46,A47,A48,A50,A59,GOBRD13:25; hence left_cell(f|(k+1),m,G) misses C by A3,A18,A19,A16 ,A40,GOBRD13:31; cell(G,i2-'1,j2) meets C by A3,A39,A38,A41,A45,A46,A47 ,A44,A58,GOBRD13:28; then right_cell(f,m,G) meets C by A3,A18,A8,A19,A16,A40 ,A46,A47,A48,A59,GOBRD13:26; hence right_cell(f|(k+1),m,G) meets C by A3,A18,A19,A16 ,A40,GOBRD13:31; end; end; hence thesis; end; suppose A60: front_left_cell(f,(len(f|k))-'1,G) misses C & front_right_cell(f,(len(f|k))-'1,G) meets C; then A61: f goes_straight (len(f|k))-'1,G by A2,A18,A8,A39,A43,Def1; now per cases by A38,A45,A46,A47,A43,A61,GOBRD13:def 8; suppose that A62: i1 = i2 & j1+1 = j2 and A63: [i2,j2+1] in Indices G & f/.(len(f|k)+1) = G*( i2,j2+1); front_left_cell(f,(len(f|k))-'1,G) = cell(G,i1-'1, j2) by A3,A39,A38,A41,A45,A46,A47,A62,GOBRD13:34; then left_cell(f,m,G) misses C by A3,A18,A8,A19,A16,A40 ,A46,A47,A60,A62,A63,GOBRD13:21; hence left_cell(f|(k+1),m,G) misses C by A3,A18,A19,A16 ,A40,GOBRD13:31; front_right_cell(f,(len(f|k))-'1,G) = cell(G,i1,j2 ) by A3,A39,A38,A41,A45,A46,A47,A62,GOBRD13:35; then right_cell(f,m,G) meets C by A3,A18,A8,A19,A16,A40 ,A46,A47,A60,A62,A63,GOBRD13:22; hence right_cell(f|(k+1),m,G) meets C by A3,A18,A19,A16 ,A40,GOBRD13:31; end; suppose that A64: i1+1 = i2 & j1 = j2 and A65: [i2+1,j2] in Indices G & f/.(len(f|k)+1) = G*( i2+1,j2); front_left_cell(f,(len(f|k))-'1,G) = cell(G,i2,j2) by A3,A39,A38,A41,A45,A46,A47,A64,GOBRD13:36; then left_cell(f,m,G) misses C by A3,A18,A8,A19,A16,A40 ,A46,A47,A60,A65,GOBRD13:23; hence left_cell(f|(k+1),m,G) misses C by A3,A18,A19,A16 ,A40,GOBRD13:31; front_right_cell(f,(len(f|k))-'1,G) = cell(G,i2,j2 -'1) by A3,A39,A38,A41,A45,A46,A47,A64,GOBRD13:37; then right_cell(f,m,G) meets C by A3,A18,A8,A19,A16,A40 ,A46,A47,A60,A65,GOBRD13:24; hence right_cell(f|(k+1),m,G) meets C by A3,A18,A19,A16 ,A40,GOBRD13:31; end; suppose that A66: i1 = i2+1 & j1 = j2 and A67: [i2-'1,j2] in Indices G & f/.(len(f|k)+1) = G* (i2-'1,j2); front_left_cell(f,(len(f|k))-'1,G) = cell(G,i2-'1, j2-'1) by A3,A39,A38,A41,A45,A46,A47,A66,GOBRD13:38; then left_cell(f,m,G) misses C by A3,A18,A8,A19,A16,A40 ,A46,A47,A48,A60,A67,GOBRD13:25; hence left_cell(f|(k+1),m,G) misses C by A3,A18,A19,A16 ,A40,GOBRD13:31; front_right_cell(f,(len(f|k))-'1,G) = cell(G,i2-'1 ,j2) by A3,A39,A38,A41,A45,A46,A47,A66,GOBRD13:39; then right_cell(f,m,G) meets C by A3,A18,A8,A19,A16,A40 ,A46,A47,A48,A60,A67,GOBRD13:26; hence right_cell(f|(k+1),m,G) meets C by A3,A18,A19,A16 ,A40,GOBRD13:31; end; suppose that A68: i1 = i2 & j1 = j2+1 and A69: [i2,j2-'1] in Indices G & f/.(len(f|k)+1) = G* (i2,j2-'1); front_left_cell(f,(len(f|k))-'1,G) = cell(G,i2,j2 -'1) by A3,A39,A38,A41,A45,A46,A47,A68,GOBRD13:40; then left_cell(f,m,G) misses C by A3,A18,A8,A19,A16,A40 ,A46,A47,A49,A60,A69,GOBRD13:27; hence left_cell(f|(k+1),m,G) misses C by A3,A18,A19,A16 ,A40,GOBRD13:31; front_right_cell(f,(len(f|k))-'1,G) = cell(G,i2-'1 ,j2-'1 ) by A3,A39,A38,A41,A45,A46,A47,A68,GOBRD13:41; then right_cell(f,m,G) meets C by A3,A18,A8,A19,A16,A40 ,A46,A47,A49,A60,A69,GOBRD13:28; hence right_cell(f|(k+1),m,G) meets C by A3,A18,A19,A16 ,A40,GOBRD13:31; end; end; hence thesis; end; suppose A70: front_left_cell(f,(len(f|k))-'1,G) meets C; then A71: f turns_left (len(f|k))-'1,G by A2,A18,A8,A39,A43,Def1; now per cases by A38,A45,A46,A47,A43,A71,GOBRD13:def 7; suppose that A72: i1 = i2 & j1+1 = j2 and A73: [i2-'1,j2] in Indices G & f/.(len(f|k)+1) = G* (i2-'1,j2); j2-'1 = j1 & cell(G,i1-'1,j1) misses C by A3,A39,A38,A41 ,A45,A46,A47,A42,A72,GOBRD13:21,NAT_D:34; then left_cell(f,m,G) misses C by A3,A18,A8,A19,A16,A40 ,A46,A47,A48,A72,A73,GOBRD13:25; hence left_cell(f|(k+1),m,G) misses C by A3,A18,A19,A16 ,A40,GOBRD13:31; front_left_cell(f,(len(f|k))-'1,G) = cell(G,i1-'1, j2) by A3,A39,A38,A41,A45,A46,A47,A72,GOBRD13:34; then right_cell(f,m,G) meets C by A3,A18,A8,A19,A16,A40 ,A46,A47,A48,A70,A72,A73,GOBRD13:26; hence right_cell(f|(k+1),m,G) meets C by A3,A18,A19,A16 ,A40,GOBRD13:31; end; suppose that A74: i1+1 = i2 & j1 = j2 and A75: [i2,j2+1] in Indices G & f/.(len(f|k)+1) = G*( i2,j2+1); i2-'1 = i1 & cell(G,i1,j1) misses C by A3,A39,A38,A41,A45 ,A46,A47,A42,A74,GOBRD13:23,NAT_D:34; then left_cell(f,m,G) misses C by A3,A18,A8,A19,A16,A40 ,A46,A47,A74,A75,GOBRD13:21; hence left_cell(f|(k+1),m,G) misses C by A3,A18,A19,A16 ,A40,GOBRD13:31; front_left_cell(f,(len(f|k))-'1,G) = cell(G,i2,j2) by A3,A39,A38,A41,A45,A46,A47,A74,GOBRD13:36; then right_cell(f,m,G) meets C by A3,A18,A8,A19,A16,A40 ,A46,A47,A70,A75,GOBRD13:22; hence right_cell(f|(k+1),m,G) meets C by A3,A18,A19,A16 ,A40,GOBRD13:31; end; suppose that A76: i1 = i2+1 & j1 = j2 and A77: [i2,j2-'1] in Indices G & f/.(len(f|k)+1) = G* (i2,j2-'1); cell(G,i2,j2-'1) misses C by A3,A39,A38,A41,A45,A46,A47 ,A42,A76,GOBRD13:25; then left_cell(f,m,G) misses C by A3,A18,A8,A19,A16,A40 ,A46,A47,A49,A77,GOBRD13:27; hence left_cell(f|(k+1),m,G) misses C by A3,A18,A19,A16 ,A40,GOBRD13:31; front_left_cell(f,(len(f|k))-'1,G) = cell(G,i2-'1, j2-'1) by A3,A39,A38,A41,A45,A46,A47,A76,GOBRD13:38; then right_cell(f,m,G) meets C by A3,A18,A8,A19,A16,A40 ,A46,A47,A49,A70,A77,GOBRD13:28; hence right_cell(f|(k+1),m,G) meets C by A3,A18,A19,A16 ,A40,GOBRD13:31; end; suppose that A78: i1 = i2 & j1 = j2+1 and A79: [i2+1,j2] in Indices G & f/.(len(f|k)+1) = G*( i2+1,j2); cell(G,i2,j2) misses C by A3,A39,A38,A41,A45,A46,A47,A42 ,A78,GOBRD13:27; then left_cell(f,m,G) misses C by A3,A18,A8,A19,A16,A40 ,A46,A47,A79,GOBRD13:23; hence left_cell(f|(k+1),m,G) misses C by A3,A18,A19,A16 ,A40,GOBRD13:31; front_left_cell(f,(len(f|k))-'1,G) = cell(G,i2,j2 -'1) by A3,A39,A38,A41,A45,A46,A47,A78,GOBRD13:40; then right_cell(f,m,G) meets C by A3,A18,A8,A19,A16,A40 ,A46,A47,A70,A79,GOBRD13:24; hence right_cell(f|(k+1),m,G) meets C by A3,A18,A19,A16 ,A40,GOBRD13:31; end; end; hence thesis; end; end; hence thesis; end; suppose m+1 <> len(f|(k+1)); then m+1 < len(f|(k+1)) by A17,XXREAL_0:1; then A80: m+1 <= len(f|k)by A8,A19,NAT_1:13; then consider i1,j1,i2,j2 being Element of NAT such that A81: [i1,j1] in Indices G and A82: (f|k)/.m = G*(i1,j1) and A83: [i2,j2] in Indices G and A84: (f|k)/.(m+1) = G*(i2,j2) and A85: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+ 1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A10,A16,JORDAN8:3; A86: left_cell(f|k,m,G) misses C & right_cell(f|k,m,G) meets C by A6,A16,A80; A87: f|(k+1) = (f|k)^<*f/.(k+1)*> by A18,FINSEQ_5:82; 1 <= m+1 by NAT_1:12; then m+1 in dom(f|k) by A80,FINSEQ_3:25; then A88: (f|(k+1))/.(m+1) = G*(i2,j2) by A84,A87,FINSEQ_4:68; m <= len(f|k) by A80,NAT_1:13; then m in dom(f|k) by A16,FINSEQ_3:25; then A89: (f|(k+1))/.m = G*(i1,j1) by A82,A87,FINSEQ_4:68; now per cases by A85; suppose A90: i1 = i2 & j1+1 = j2; then left_cell(f|k,m,G) = cell(G,i1-'1,j1) & right_cell(f|k ,m,G) = cell(G,i1,j1) by A10,A16,A80,A81,A82,A83,A84,GOBRD13:21,22; hence thesis by A11,A16,A17,A81,A83,A86,A89,A88,A90, GOBRD13:21,22; end; suppose A91: i1+1 = i2 & j1 = j2; then left_cell(f|k,m,G) = cell(G,i1,j1) & right_cell(f|k,m, G) = cell(G,i1,j1-'1) by A10,A16,A80,A81,A82,A83,A84,GOBRD13:23,24; hence thesis by A11,A16,A17,A81,A83,A86,A89,A88,A91, GOBRD13:23,24; end; suppose A92: i1 = i2+1 & j1 = j2; then left_cell(f|k,m,G) = cell(G,i2,j2-'1) & right_cell(f|k ,m,G) = cell(G,i2,j2) by A10,A16,A80,A81,A82,A83,A84,GOBRD13:25,26; hence thesis by A11,A16,A17,A81,A83,A86,A89,A88,A92, GOBRD13:25,26; end; suppose A93: i1 = i2 & j1 = j2+1; then left_cell(f|k,m,G) = cell(G,i2,j2) & right_cell(f|k,m, G) = cell(G,i1-'1,j2) by A10,A16,A80,A81,A82,A83,A84,GOBRD13:27,28; hence thesis by A11,A16,A17,A81,A83,A86,A89,A88,A93, GOBRD13:27,28; end; end; hence thesis; end; end; hence thesis; end; end; hence thesis; end; end; A94: f|len f = f by FINSEQ_1:58; A95: P[0] by CARD_1:27; for k holds P[k] from NAT_1:sch 1(A95,A5); hence thesis by A94; end; theorem C is connected implies N-min L~Cage(C,n) = (Cage(C,n))/.1 proof set f = Cage(C,n); assume A1: C is connected; then A2: for k st 1 <= k & k+1 <= len f holds left_cell(f,k,Gauge(C,n)) misses C & right_cell(f,k,Gauge(C,n)) meets C by Th31; f is_sequence_on Gauge(C,n) & ex i st 1 <= i & i+1 <= len Gauge(C,n ) & f/.1 = Gauge(C,n)*(i,width Gauge(C,n)) & f/.2 = Gauge(C,n)*(i+1,width Gauge(C,n )) & N-min C in cell(Gauge( C,n),i,width Gauge(C,n)-'1) & N-min C <> Gauge(C,n) *(i, width Gauge(C,n)-'1) by A1,Def1; hence thesis by A2,Th30; end;