:: One-Side Limits of a Real Function at a Point
:: by Jaros{\l}aw Kotowicz
::
:: Copyright (c) 1990-2017 Association of Mizar Users

Lm1: for r, g, r1 being Real st 0 < g & r <= r1 holds
( r - g < r1 & r < r1 + g )

proof end;

Lm2: for seq being Real_Sequence
for f1, f2 being PartFunc of REAL,REAL
for X being Subset of REAL st rng seq c= (dom (f1 (#) f2)) /\ X holds
( rng seq c= dom (f1 (#) f2) & rng seq c= X & dom (f1 (#) f2) = (dom f1) /\ (dom f2) & rng seq c= dom f1 & rng seq c= dom f2 & rng seq c= (dom f1) /\ X & rng seq c= (dom f2) /\ X )

proof end;

Lm3: for r being Real
for n being Nat holds
( r - (1 / (n + 1)) < r & r < r + (1 / (n + 1)) )

proof end;

Lm4: for seq being Real_Sequence
for f1, f2 being PartFunc of REAL,REAL
for X being Subset of REAL st rng seq c= (dom (f1 + f2)) /\ X holds
( rng seq c= dom (f1 + f2) & rng seq c= X & dom (f1 + f2) = (dom f1) /\ (dom f2) & rng seq c= (dom f1) /\ X & rng seq c= (dom f2) /\ X )

proof end;

theorem Th1: :: LIMFUNC2:1
for r being Real
for seq being Real_Sequence st seq is convergent & r < lim seq holds
ex n being Nat st
for k being Nat st n <= k holds
r < seq . k
proof end;

theorem Th2: :: LIMFUNC2:2
for r being Real
for seq being Real_Sequence st seq is convergent & lim seq < r holds
ex n being Nat st
for k being Nat st n <= k holds
seq . k < r
proof end;

Lm5: for x0 being Real
for seq being Real_Sequence
for f being PartFunc of REAL,REAL st ( for g1 being Real ex r being Real st
( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds
f . r1 < g1 ) ) ) & seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ () holds
f /* seq is divergent_to-infty

proof end;

theorem Th3: :: LIMFUNC2:3
for r1, r2 being Real
for f being PartFunc of REAL,REAL st 0 < r2 & ].(r1 - r2),r1.[ c= dom f holds
for r being Real st r < r1 holds
ex g being Real st
( r < g & g < r1 & g in dom f )
proof end;

theorem Th4: :: LIMFUNC2:4
for r1, r2 being Real
for f being PartFunc of REAL,REAL st 0 < r2 & ].r1,(r1 + r2).[ c= dom f holds
for r being Real st r1 < r holds
ex g being Real st
( g < r & r1 < g & g in dom f )
proof end;

theorem Th5: :: LIMFUNC2:5
for x0 being Real
for seq being Real_Sequence
for f being PartFunc of REAL,REAL st ( for n being Nat holds
( x0 - (1 / (n + 1)) < seq . n & seq . n < x0 & seq . n in dom f ) ) holds
( seq is convergent & lim seq = x0 & rng seq c= dom f & rng seq c= (dom f) /\ () )
proof end;

theorem Th6: :: LIMFUNC2:6
for x0 being Real
for seq being Real_Sequence
for f being PartFunc of REAL,REAL st ( for n being Nat holds
( x0 < seq . n & seq . n < x0 + (1 / (n + 1)) & seq . n in dom f ) ) holds
( seq is convergent & lim seq = x0 & rng seq c= dom f & rng seq c= (dom f) /\ () )
proof end;

definition
let f be PartFunc of REAL,REAL;
let x0 be Real;
pred f is_left_convergent_in x0 means :: LIMFUNC2:def 1
( ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom f ) ) & ex g being Real st
for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ () holds
( f /* seq is convergent & lim (f /* seq) = g ) );
pred f is_left_divergent_to+infty_in x0 means :: LIMFUNC2:def 2
( ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom f ) ) & ( for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ () holds
f /* seq is divergent_to+infty ) );
pred f is_left_divergent_to-infty_in x0 means :: LIMFUNC2:def 3
( ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom f ) ) & ( for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ () holds
f /* seq is divergent_to-infty ) );
pred f is_right_convergent_in x0 means :: LIMFUNC2:def 4
( ( for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom f ) ) & ex g being Real st
for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ () holds
( f /* seq is convergent & lim (f /* seq) = g ) );
pred f is_right_divergent_to+infty_in x0 means :: LIMFUNC2:def 5
( ( for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom f ) ) & ( for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ () holds
f /* seq is divergent_to+infty ) );
pred f is_right_divergent_to-infty_in x0 means :: LIMFUNC2:def 6
( ( for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom f ) ) & ( for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ () holds
f /* seq is divergent_to-infty ) );
end;

:: deftheorem defines is_left_convergent_in LIMFUNC2:def 1 :
for f being PartFunc of REAL,REAL
for x0 being Real holds
( f is_left_convergent_in x0 iff ( ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom f ) ) & ex g being Real st
for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ () holds
( f /* seq is convergent & lim (f /* seq) = g ) ) );

:: deftheorem defines is_left_divergent_to+infty_in LIMFUNC2:def 2 :
for f being PartFunc of REAL,REAL
for x0 being Real holds
( f is_left_divergent_to+infty_in x0 iff ( ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom f ) ) & ( for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ () holds
f /* seq is divergent_to+infty ) ) );

:: deftheorem defines is_left_divergent_to-infty_in LIMFUNC2:def 3 :
for f being PartFunc of REAL,REAL
for x0 being Real holds
( f is_left_divergent_to-infty_in x0 iff ( ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom f ) ) & ( for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ () holds
f /* seq is divergent_to-infty ) ) );

:: deftheorem defines is_right_convergent_in LIMFUNC2:def 4 :
for f being PartFunc of REAL,REAL
for x0 being Real holds
( f is_right_convergent_in x0 iff ( ( for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom f ) ) & ex g being Real st
for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ () holds
( f /* seq is convergent & lim (f /* seq) = g ) ) );

:: deftheorem defines is_right_divergent_to+infty_in LIMFUNC2:def 5 :
for f being PartFunc of REAL,REAL
for x0 being Real holds
( f is_right_divergent_to+infty_in x0 iff ( ( for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom f ) ) & ( for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ () holds
f /* seq is divergent_to+infty ) ) );

:: deftheorem defines is_right_divergent_to-infty_in LIMFUNC2:def 6 :
for f being PartFunc of REAL,REAL
for x0 being Real holds
( f is_right_divergent_to-infty_in x0 iff ( ( for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom f ) ) & ( for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ () holds
f /* seq is divergent_to-infty ) ) );

theorem :: LIMFUNC2:7
for x0 being Real
for f being PartFunc of REAL,REAL holds
( f is_left_convergent_in x0 iff ( ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom f ) ) & ex g being Real st
for g1 being Real st 0 < g1 holds
ex r being Real st
( r < x0 & ( for r1 being Real st r < r1 & r1 < x0 & r1 in dom f holds
|.((f . r1) - g).| < g1 ) ) ) )
proof end;

theorem :: LIMFUNC2:8
for x0 being Real
for f being PartFunc of REAL,REAL holds
( f is_left_divergent_to+infty_in x0 iff ( ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom f ) ) & ( for g1 being Real ex r being Real st
( r < x0 & ( for r1 being Real st r < r1 & r1 < x0 & r1 in dom f holds
g1 < f . r1 ) ) ) ) )
proof end;

theorem :: LIMFUNC2:9
for x0 being Real
for f being PartFunc of REAL,REAL holds
( f is_left_divergent_to-infty_in x0 iff ( ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom f ) ) & ( for g1 being Real ex r being Real st
( r < x0 & ( for r1 being Real st r < r1 & r1 < x0 & r1 in dom f holds
f . r1 < g1 ) ) ) ) )
proof end;

theorem :: LIMFUNC2:10
for x0 being Real
for f being PartFunc of REAL,REAL holds
( f is_right_convergent_in x0 iff ( ( for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom f ) ) & ex g being Real st
for g1 being Real st 0 < g1 holds
ex r being Real st
( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds
|.((f . r1) - g).| < g1 ) ) ) )
proof end;

theorem :: LIMFUNC2:11
for x0 being Real
for f being PartFunc of REAL,REAL holds
( f is_right_divergent_to+infty_in x0 iff ( ( for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom f ) ) & ( for g1 being Real ex r being Real st
( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds
g1 < f . r1 ) ) ) ) )
proof end;

theorem :: LIMFUNC2:12
for x0 being Real
for f being PartFunc of REAL,REAL holds
( f is_right_divergent_to-infty_in x0 iff ( ( for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom f ) ) & ( for g1 being Real ex r being Real st
( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds
f . r1 < g1 ) ) ) ) )
proof end;

theorem :: LIMFUNC2:13
for x0 being Real
for f1, f2 being PartFunc of REAL,REAL st f1 is_left_divergent_to+infty_in x0 & f2 is_left_divergent_to+infty_in x0 & ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in (dom f1) /\ (dom f2) ) ) holds
( f1 + f2 is_left_divergent_to+infty_in x0 & f1 (#) f2 is_left_divergent_to+infty_in x0 )
proof end;

theorem :: LIMFUNC2:14
for x0 being Real
for f1, f2 being PartFunc of REAL,REAL st f1 is_left_divergent_to-infty_in x0 & f2 is_left_divergent_to-infty_in x0 & ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in (dom f1) /\ (dom f2) ) ) holds
( f1 + f2 is_left_divergent_to-infty_in x0 & f1 (#) f2 is_left_divergent_to+infty_in x0 )
proof end;

theorem :: LIMFUNC2:15
for x0 being Real
for f1, f2 being PartFunc of REAL,REAL st f1 is_right_divergent_to+infty_in x0 & f2 is_right_divergent_to+infty_in x0 & ( for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in (dom f1) /\ (dom f2) ) ) holds
( f1 + f2 is_right_divergent_to+infty_in x0 & f1 (#) f2 is_right_divergent_to+infty_in x0 )
proof end;

theorem :: LIMFUNC2:16
for x0 being Real
for f1, f2 being PartFunc of REAL,REAL st f1 is_right_divergent_to-infty_in x0 & f2 is_right_divergent_to-infty_in x0 & ( for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in (dom f1) /\ (dom f2) ) ) holds
( f1 + f2 is_right_divergent_to-infty_in x0 & f1 (#) f2 is_right_divergent_to+infty_in x0 )
proof end;

theorem :: LIMFUNC2:17
for x0 being Real
for f1, f2 being PartFunc of REAL,REAL st f1 is_left_divergent_to+infty_in x0 & ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom (f1 + f2) ) ) & ex r being Real st
( 0 < r & f2 | ].(x0 - r),x0.[ is bounded_below ) holds
f1 + f2 is_left_divergent_to+infty_in x0
proof end;

theorem :: LIMFUNC2:18
for x0 being Real
for f1, f2 being PartFunc of REAL,REAL st f1 is_left_divergent_to+infty_in x0 & ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom (f1 (#) f2) ) ) & ex r, r1 being Real st
( 0 < r & 0 < r1 & ( for g being Real st g in (dom f2) /\ ].(x0 - r),x0.[ holds
r1 <= f2 . g ) ) holds
f1 (#) f2 is_left_divergent_to+infty_in x0
proof end;

theorem :: LIMFUNC2:19
for x0 being Real
for f1, f2 being PartFunc of REAL,REAL st f1 is_right_divergent_to+infty_in x0 & ( for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom (f1 + f2) ) ) & ex r being Real st
( 0 < r & f2 | ].x0,(x0 + r).[ is bounded_below ) holds
f1 + f2 is_right_divergent_to+infty_in x0
proof end;

theorem :: LIMFUNC2:20
for x0 being Real
for f1, f2 being PartFunc of REAL,REAL st f1 is_right_divergent_to+infty_in x0 & ( for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom (f1 (#) f2) ) ) & ex r, r1 being Real st
( 0 < r & 0 < r1 & ( for g being Real st g in (dom f2) /\ ].x0,(x0 + r).[ holds
r1 <= f2 . g ) ) holds
f1 (#) f2 is_right_divergent_to+infty_in x0
proof end;

theorem :: LIMFUNC2:21
for r, x0 being Real
for f being PartFunc of REAL,REAL holds
( ( f is_left_divergent_to+infty_in x0 & r > 0 implies r (#) f is_left_divergent_to+infty_in x0 ) & ( f is_left_divergent_to+infty_in x0 & r < 0 implies r (#) f is_left_divergent_to-infty_in x0 ) & ( f is_left_divergent_to-infty_in x0 & r > 0 implies r (#) f is_left_divergent_to-infty_in x0 ) & ( f is_left_divergent_to-infty_in x0 & r < 0 implies r (#) f is_left_divergent_to+infty_in x0 ) )
proof end;

theorem :: LIMFUNC2:22
for r, x0 being Real
for f being PartFunc of REAL,REAL holds
( ( f is_right_divergent_to+infty_in x0 & r > 0 implies r (#) f is_right_divergent_to+infty_in x0 ) & ( f is_right_divergent_to+infty_in x0 & r < 0 implies r (#) f is_right_divergent_to-infty_in x0 ) & ( f is_right_divergent_to-infty_in x0 & r > 0 implies r (#) f is_right_divergent_to-infty_in x0 ) & ( f is_right_divergent_to-infty_in x0 & r < 0 implies r (#) f is_right_divergent_to+infty_in x0 ) )
proof end;

theorem :: LIMFUNC2:23
for x0 being Real
for f being PartFunc of REAL,REAL st ( f is_left_divergent_to+infty_in x0 or f is_left_divergent_to-infty_in x0 ) holds
abs f is_left_divergent_to+infty_in x0
proof end;

theorem :: LIMFUNC2:24
for x0 being Real
for f being PartFunc of REAL,REAL st ( f is_right_divergent_to+infty_in x0 or f is_right_divergent_to-infty_in x0 ) holds
abs f is_right_divergent_to+infty_in x0
proof end;

theorem Th25: :: LIMFUNC2:25
for x0 being Real
for f being PartFunc of REAL,REAL st ex r being Real st
( f | ].(x0 - r),x0.[ is non-decreasing & not f | ].(x0 - r),x0.[ is bounded_above ) & ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom f ) ) holds
f is_left_divergent_to+infty_in x0
proof end;

theorem :: LIMFUNC2:26
for x0 being Real
for f being PartFunc of REAL,REAL st ex r being Real st
( 0 < r & f | ].(x0 - r),x0.[ is increasing & not f | ].(x0 - r),x0.[ is bounded_above ) & ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom f ) ) holds
f is_left_divergent_to+infty_in x0 by Th25;

theorem Th27: :: LIMFUNC2:27
for x0 being Real
for f being PartFunc of REAL,REAL st ex r being Real st
( f | ].(x0 - r),x0.[ is non-increasing & not f | ].(x0 - r),x0.[ is bounded_below ) & ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom f ) ) holds
f is_left_divergent_to-infty_in x0
proof end;

theorem :: LIMFUNC2:28
for x0 being Real
for f being PartFunc of REAL,REAL st ex r being Real st
( 0 < r & f | ].(x0 - r),x0.[ is decreasing & not f | ].(x0 - r),x0.[ is bounded_below ) & ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom f ) ) holds
f is_left_divergent_to-infty_in x0 by Th27;

theorem Th29: :: LIMFUNC2:29
for x0 being Real
for f being PartFunc of REAL,REAL st ex r being Real st
( f | ].x0,(x0 + r).[ is non-increasing & not f | ].x0,(x0 + r).[ is bounded_above ) & ( for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom f ) ) holds
f is_right_divergent_to+infty_in x0
proof end;

theorem :: LIMFUNC2:30
for x0 being Real
for f being PartFunc of REAL,REAL st ex r being Real st
( 0 < r & f | ].x0,(x0 + r).[ is decreasing & not f | ].x0,(x0 + r).[ is bounded_above ) & ( for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom f ) ) holds
f is_right_divergent_to+infty_in x0 by Th29;

theorem Th31: :: LIMFUNC2:31
for x0 being Real
for f being PartFunc of REAL,REAL st ex r being Real st
( f | ].x0,(x0 + r).[ is non-decreasing & not f | ].x0,(x0 + r).[ is bounded_below ) & ( for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom f ) ) holds
f is_right_divergent_to-infty_in x0
proof end;

theorem :: LIMFUNC2:32
for x0 being Real
for f being PartFunc of REAL,REAL st ex r being Real st
( 0 < r & f | ].x0,(x0 + r).[ is increasing & not f | ].x0,(x0 + r).[ is bounded_below ) & ( for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom f ) ) holds
f is_right_divergent_to-infty_in x0 by Th31;

theorem Th33: :: LIMFUNC2:33
for x0 being Real
for f, f1 being PartFunc of REAL,REAL st f1 is_left_divergent_to+infty_in x0 & ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom f ) ) & ex r being Real st
( 0 < r & (dom f) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ & ( for g being Real st g in (dom f) /\ ].(x0 - r),x0.[ holds
f1 . g <= f . g ) ) holds
f is_left_divergent_to+infty_in x0
proof end;

theorem Th34: :: LIMFUNC2:34
for x0 being Real
for f, f1 being PartFunc of REAL,REAL st f1 is_left_divergent_to-infty_in x0 & ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom f ) ) & ex r being Real st
( 0 < r & (dom f) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ & ( for g being Real st g in (dom f) /\ ].(x0 - r),x0.[ holds
f . g <= f1 . g ) ) holds
f is_left_divergent_to-infty_in x0
proof end;

theorem Th35: :: LIMFUNC2:35
for x0 being Real
for f, f1 being PartFunc of REAL,REAL st f1 is_right_divergent_to+infty_in x0 & ( for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom f ) ) & ex r being Real st
( 0 < r & (dom f) /\ ].x0,(x0 + r).[ c= (dom f1) /\ ].x0,(x0 + r).[ & ( for g being Real st g in (dom f) /\ ].x0,(x0 + r).[ holds
f1 . g <= f . g ) ) holds
f is_right_divergent_to+infty_in x0
proof end;

theorem Th36: :: LIMFUNC2:36
for x0 being Real
for f, f1 being PartFunc of REAL,REAL st f1 is_right_divergent_to-infty_in x0 & ( for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom f ) ) & ex r being Real st
( 0 < r & (dom f) /\ ].x0,(x0 + r).[ c= (dom f1) /\ ].x0,(x0 + r).[ & ( for g being Real st g in (dom f) /\ ].x0,(x0 + r).[ holds
f . g <= f1 . g ) ) holds
f is_right_divergent_to-infty_in x0
proof end;

theorem :: LIMFUNC2:37
for x0 being Real
for f, f1 being PartFunc of REAL,REAL st f1 is_left_divergent_to+infty_in x0 & ex r being Real st
( 0 < r & ].(x0 - r),x0.[ c= (dom f) /\ (dom f1) & ( for g being Real st g in ].(x0 - r),x0.[ holds
f1 . g <= f . g ) ) holds
f is_left_divergent_to+infty_in x0
proof end;

theorem :: LIMFUNC2:38
for x0 being Real
for f, f1 being PartFunc of REAL,REAL st f1 is_left_divergent_to-infty_in x0 & ex r being Real st
( 0 < r & ].(x0 - r),x0.[ c= (dom f) /\ (dom f1) & ( for g being Real st g in ].(x0 - r),x0.[ holds
f . g <= f1 . g ) ) holds
f is_left_divergent_to-infty_in x0
proof end;

theorem :: LIMFUNC2:39
for x0 being Real
for f, f1 being PartFunc of REAL,REAL st f1 is_right_divergent_to+infty_in x0 & ex r being Real st
( 0 < r & ].x0,(x0 + r).[ c= (dom f) /\ (dom f1) & ( for g being Real st g in ].x0,(x0 + r).[ holds
f1 . g <= f . g ) ) holds
f is_right_divergent_to+infty_in x0
proof end;

theorem :: LIMFUNC2:40
for x0 being Real
for f, f1 being PartFunc of REAL,REAL st f1 is_right_divergent_to-infty_in x0 & ex r being Real st
( 0 < r & ].x0,(x0 + r).[ c= (dom f) /\ (dom f1) & ( for g being Real st g in ].x0,(x0 + r).[ holds
f . g <= f1 . g ) ) holds
f is_right_divergent_to-infty_in x0
proof end;

definition
let f be PartFunc of REAL,REAL;
let x0 be Real;
assume A1: f is_left_convergent_in x0 ;
func lim_left (f,x0) -> Real means :Def7: :: LIMFUNC2:def 7
for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ () holds
( f /* seq is convergent & lim (f /* seq) = it );
existence
ex b1 being Real st
for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ () holds
( f /* seq is convergent & lim (f /* seq) = b1 )
by A1;
uniqueness
for b1, b2 being Real st ( for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ () holds
( f /* seq is convergent & lim (f /* seq) = b1 ) ) & ( for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ () holds
( f /* seq is convergent & lim (f /* seq) = b2 ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def7 defines lim_left LIMFUNC2:def 7 :
for f being PartFunc of REAL,REAL
for x0 being Real st f is_left_convergent_in x0 holds
for b3 being Real holds
( b3 = lim_left (f,x0) iff for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ () holds
( f /* seq is convergent & lim (f /* seq) = b3 ) );

definition
let f be PartFunc of REAL,REAL;
let x0 be Real;
assume A1: f is_right_convergent_in x0 ;
func lim_right (f,x0) -> Real means :Def8: :: LIMFUNC2:def 8
for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ () holds
( f /* seq is convergent & lim (f /* seq) = it );
existence
ex b1 being Real st
for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ () holds
( f /* seq is convergent & lim (f /* seq) = b1 )
by A1;
uniqueness
for b1, b2 being Real st ( for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ () holds
( f /* seq is convergent & lim (f /* seq) = b1 ) ) & ( for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ () holds
( f /* seq is convergent & lim (f /* seq) = b2 ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def8 defines lim_right LIMFUNC2:def 8 :
for f being PartFunc of REAL,REAL
for x0 being Real st f is_right_convergent_in x0 holds
for b3 being Real holds
( b3 = lim_right (f,x0) iff for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ () holds
( f /* seq is convergent & lim (f /* seq) = b3 ) );

theorem :: LIMFUNC2:41
for g, x0 being Real
for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 holds
( lim_left (f,x0) = g iff for g1 being Real st 0 < g1 holds
ex r being Real st
( r < x0 & ( for r1 being Real st r < r1 & r1 < x0 & r1 in dom f holds
|.((f . r1) - g).| < g1 ) ) )
proof end;

theorem :: LIMFUNC2:42
for g, x0 being Real
for f being PartFunc of REAL,REAL st f is_right_convergent_in x0 holds
( lim_right (f,x0) = g iff for g1 being Real st 0 < g1 holds
ex r being Real st
( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds
|.((f . r1) - g).| < g1 ) ) )
proof end;

theorem Th43: :: LIMFUNC2:43
for r, x0 being Real
for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 holds
( r (#) f is_left_convergent_in x0 & lim_left ((r (#) f),x0) = r * (lim_left (f,x0)) )
proof end;

theorem Th44: :: LIMFUNC2:44
for x0 being Real
for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 holds
( - f is_left_convergent_in x0 & lim_left ((- f),x0) = - (lim_left (f,x0)) )
proof end;

theorem Th45: :: LIMFUNC2:45
for x0 being Real
for f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom (f1 + f2) ) ) holds
( f1 + f2 is_left_convergent_in x0 & lim_left ((f1 + f2),x0) = (lim_left (f1,x0)) + (lim_left (f2,x0)) )
proof end;

theorem :: LIMFUNC2:46
for x0 being Real
for f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom (f1 - f2) ) ) holds
( f1 - f2 is_left_convergent_in x0 & lim_left ((f1 - f2),x0) = (lim_left (f1,x0)) - (lim_left (f2,x0)) )
proof end;

theorem :: LIMFUNC2:47
for x0 being Real
for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 & f " = {} & lim_left (f,x0) <> 0 holds
( f ^ is_left_convergent_in x0 & lim_left ((f ^),x0) = (lim_left (f,x0)) " )
proof end;

theorem :: LIMFUNC2:48
for x0 being Real
for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 holds
( abs f is_left_convergent_in x0 & lim_left ((abs f),x0) = |.(lim_left (f,x0)).| )
proof end;

theorem Th49: :: LIMFUNC2:49
for x0 being Real
for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 & lim_left (f,x0) <> 0 & ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom f & f . g <> 0 ) ) holds
( f ^ is_left_convergent_in x0 & lim_left ((f ^),x0) = (lim_left (f,x0)) " )
proof end;

theorem Th50: :: LIMFUNC2:50
for x0 being Real
for f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom (f1 (#) f2) ) ) holds
( f1 (#) f2 is_left_convergent_in x0 & lim_left ((f1 (#) f2),x0) = (lim_left (f1,x0)) * (lim_left (f2,x0)) )
proof end;

theorem :: LIMFUNC2:51
for x0 being Real
for f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & lim_left (f2,x0) <> 0 & ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom (f1 / f2) ) ) holds
( f1 / f2 is_left_convergent_in x0 & lim_left ((f1 / f2),x0) = (lim_left (f1,x0)) / (lim_left (f2,x0)) )
proof end;

theorem Th52: :: LIMFUNC2:52
for r, x0 being Real
for f being PartFunc of REAL,REAL st f is_right_convergent_in x0 holds
( r (#) f is_right_convergent_in x0 & lim_right ((r (#) f),x0) = r * (lim_right (f,x0)) )
proof end;

theorem Th53: :: LIMFUNC2:53
for x0 being Real
for f being PartFunc of REAL,REAL st f is_right_convergent_in x0 holds
( - f is_right_convergent_in x0 & lim_right ((- f),x0) = - (lim_right (f,x0)) )
proof end;

theorem Th54: :: LIMFUNC2:54
for x0 being Real
for f1, f2 being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & f2 is_right_convergent_in x0 & ( for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom (f1 + f2) ) ) holds
( f1 + f2 is_right_convergent_in x0 & lim_right ((f1 + f2),x0) = (lim_right (f1,x0)) + (lim_right (f2,x0)) )
proof end;

theorem :: LIMFUNC2:55
for x0 being Real
for f1, f2 being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & f2 is_right_convergent_in x0 & ( for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom (f1 - f2) ) ) holds
( f1 - f2 is_right_convergent_in x0 & lim_right ((f1 - f2),x0) = (lim_right (f1,x0)) - (lim_right (f2,x0)) )
proof end;

theorem :: LIMFUNC2:56
for x0 being Real
for f being PartFunc of REAL,REAL st f is_right_convergent_in x0 & f " = {} & lim_right (f,x0) <> 0 holds
( f ^ is_right_convergent_in x0 & lim_right ((f ^),x0) = (lim_right (f,x0)) " )
proof end;

theorem :: LIMFUNC2:57
for x0 being Real
for f being PartFunc of REAL,REAL st f is_right_convergent_in x0 holds
( abs f is_right_convergent_in x0 & lim_right ((abs f),x0) = |.(lim_right (f,x0)).| )
proof end;

theorem Th58: :: LIMFUNC2:58
for x0 being Real
for f being PartFunc of REAL,REAL st f is_right_convergent_in x0 & lim_right (f,x0) <> 0 & ( for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom f & f . g <> 0 ) ) holds
( f ^ is_right_convergent_in x0 & lim_right ((f ^),x0) = (lim_right (f,x0)) " )
proof end;

theorem Th59: :: LIMFUNC2:59
for x0 being Real
for f1, f2 being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & f2 is_right_convergent_in x0 & ( for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom (f1 (#) f2) ) ) holds
( f1 (#) f2 is_right_convergent_in x0 & lim_right ((f1 (#) f2),x0) = (lim_right (f1,x0)) * (lim_right (f2,x0)) )
proof end;

theorem :: LIMFUNC2:60
for x0 being Real
for f1, f2 being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & f2 is_right_convergent_in x0 & lim_right (f2,x0) <> 0 & ( for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom (f1 / f2) ) ) holds
( f1 / f2 is_right_convergent_in x0 & lim_right ((f1 / f2),x0) = (lim_right (f1,x0)) / (lim_right (f2,x0)) )
proof end;

theorem :: LIMFUNC2:61
for x0 being Real
for f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & lim_left (f1,x0) = 0 & ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom (f1 (#) f2) ) ) & ex r being Real st
( 0 < r & f2 | ].(x0 - r),x0.[ is bounded ) holds
( f1 (#) f2 is_left_convergent_in x0 & lim_left ((f1 (#) f2),x0) = 0 )
proof end;

theorem :: LIMFUNC2:62
for x0 being Real
for f1, f2 being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & lim_right (f1,x0) = 0 & ( for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom (f1 (#) f2) ) ) & ex r being Real st
( 0 < r & f2 | ].x0,(x0 + r).[ is bounded ) holds
( f1 (#) f2 is_right_convergent_in x0 & lim_right ((f1 (#) f2),x0) = 0 )
proof end;

theorem Th63: :: LIMFUNC2:63
for x0 being Real
for f, f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & lim_left (f1,x0) = lim_left (f2,x0) & ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom f ) ) & ex r being Real st
( 0 < r & ( for g being Real st g in (dom f) /\ ].(x0 - r),x0.[ holds
( f1 . g <= f . g & f . g <= f2 . g ) ) & ( ( (dom f1) /\ ].(x0 - r),x0.[ c= (dom f2) /\ ].(x0 - r),x0.[ & (dom f) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ ) or ( (dom f2) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ & (dom f) /\ ].(x0 - r),x0.[ c= (dom f2) /\ ].(x0 - r),x0.[ ) ) ) holds
( f is_left_convergent_in x0 & lim_left (f,x0) = lim_left (f1,x0) )
proof end;

theorem :: LIMFUNC2:64
for x0 being Real
for f, f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & lim_left (f1,x0) = lim_left (f2,x0) & ex r being Real st
( 0 < r & ].(x0 - r),x0.[ c= ((dom f1) /\ (dom f2)) /\ (dom f) & ( for g being Real st g in ].(x0 - r),x0.[ holds
( f1 . g <= f . g & f . g <= f2 . g ) ) ) holds
( f is_left_convergent_in x0 & lim_left (f,x0) = lim_left (f1,x0) )
proof end;

theorem Th65: :: LIMFUNC2:65
for x0 being Real
for f, f1, f2 being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & f2 is_right_convergent_in x0 & lim_right (f1,x0) = lim_right (f2,x0) & ( for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom f ) ) & ex r being Real st
( 0 < r & ( for g being Real st g in (dom f) /\ ].x0,(x0 + r).[ holds
( f1 . g <= f . g & f . g <= f2 . g ) ) & ( ( (dom f1) /\ ].x0,(x0 + r).[ c= (dom f2) /\ ].x0,(x0 + r).[ & (dom f) /\ ].x0,(x0 + r).[ c= (dom f1) /\ ].x0,(x0 + r).[ ) or ( (dom f2) /\ ].x0,(x0 + r).[ c= (dom f1) /\ ].x0,(x0 + r).[ & (dom f) /\ ].x0,(x0 + r).[ c= (dom f2) /\ ].x0,(x0 + r).[ ) ) ) holds
( f is_right_convergent_in x0 & lim_right (f,x0) = lim_right (f1,x0) )
proof end;

theorem :: LIMFUNC2:66
for x0 being Real
for f, f1, f2 being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & f2 is_right_convergent_in x0 & lim_right (f1,x0) = lim_right (f2,x0) & ex r being Real st
( 0 < r & ].x0,(x0 + r).[ c= ((dom f1) /\ (dom f2)) /\ (dom f) & ( for g being Real st g in ].x0,(x0 + r).[ holds
( f1 . g <= f . g & f . g <= f2 . g ) ) ) holds
( f is_right_convergent_in x0 & lim_right (f,x0) = lim_right (f1,x0) )
proof end;

theorem :: LIMFUNC2:67
for x0 being Real
for f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & ex r being Real st
( 0 < r & ( ( (dom f1) /\ ].(x0 - r),x0.[ c= (dom f2) /\ ].(x0 - r),x0.[ & ( for g being Real st g in (dom f1) /\ ].(x0 - r),x0.[ holds
f1 . g <= f2 . g ) ) or ( (dom f2) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ & ( for g being Real st g in (dom f2) /\ ].(x0 - r),x0.[ holds
f1 . g <= f2 . g ) ) ) ) holds
lim_left (f1,x0) <= lim_left (f2,x0)
proof end;

theorem :: LIMFUNC2:68
for x0 being Real
for f1, f2 being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & f2 is_right_convergent_in x0 & ex r being Real st
( 0 < r & ( ( (dom f1) /\ ].x0,(x0 + r).[ c= (dom f2) /\ ].x0,(x0 + r).[ & ( for g being Real st g in (dom f1) /\ ].x0,(x0 + r).[ holds
f1 . g <= f2 . g ) ) or ( (dom f2) /\ ].x0,(x0 + r).[ c= (dom f1) /\ ].x0,(x0 + r).[ & ( for g being Real st g in (dom f2) /\ ].x0,(x0 + r).[ holds
f1 . g <= f2 . g ) ) ) ) holds
lim_right (f1,x0) <= lim_right (f2,x0)
proof end;

theorem :: LIMFUNC2:69
for x0 being Real
for f being PartFunc of REAL,REAL st ( f is_left_divergent_to+infty_in x0 or f is_left_divergent_to-infty_in x0 ) & ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom f & f . g <> 0 ) ) holds
( f ^ is_left_convergent_in x0 & lim_left ((f ^),x0) = 0 )
proof end;

theorem :: LIMFUNC2:70
for x0 being Real
for f being PartFunc of REAL,REAL st ( f is_right_divergent_to+infty_in x0 or f is_right_divergent_to-infty_in x0 ) & ( for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom f & f . g <> 0 ) ) holds
( f ^ is_right_convergent_in x0 & lim_right ((f ^),x0) = 0 )
proof end;

theorem :: LIMFUNC2:71
for x0 being Real
for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 & lim_left (f,x0) = 0 & ex r being Real st
( 0 < r & ( for g being Real st g in (dom f) /\ ].(x0 - r),x0.[ holds
0 < f . g ) ) holds
f ^ is_left_divergent_to+infty_in x0
proof end;

theorem :: LIMFUNC2:72
for x0 being Real
for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 & lim_left (f,x0) = 0 & ex r being Real st
( 0 < r & ( for g being Real st g in (dom f) /\ ].(x0 - r),x0.[ holds
f . g < 0 ) ) holds
f ^ is_left_divergent_to-infty_in x0
proof end;

theorem :: LIMFUNC2:73
for x0 being Real
for f being PartFunc of REAL,REAL st f is_right_convergent_in x0 & lim_right (f,x0) = 0 & ex r being Real st
( 0 < r & ( for g being Real st g in (dom f) /\ ].x0,(x0 + r).[ holds
0 < f . g ) ) holds
f ^ is_right_divergent_to+infty_in x0
proof end;

theorem :: LIMFUNC2:74
for x0 being Real
for f being PartFunc of REAL,REAL st f is_right_convergent_in x0 & lim_right (f,x0) = 0 & ex r being Real st
( 0 < r & ( for g being Real st g in (dom f) /\ ].x0,(x0 + r).[ holds
f . g < 0 ) ) holds
f ^ is_right_divergent_to-infty_in x0
proof end;

theorem :: LIMFUNC2:75
for x0 being Real
for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 & lim_left (f,x0) = 0 & ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom f & f . g <> 0 ) ) & ex r being Real st
( 0 < r & ( for g being Real st g in (dom f) /\ ].(x0 - r),x0.[ holds
0 <= f . g ) ) holds
f ^ is_left_divergent_to+infty_in x0
proof end;

theorem :: LIMFUNC2:76
for x0 being Real
for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 & lim_left (f,x0) = 0 & ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom f & f . g <> 0 ) ) & ex r being Real st
( 0 < r & ( for g being Real st g in (dom f) /\ ].(x0 - r),x0.[ holds
f . g <= 0 ) ) holds
f ^ is_left_divergent_to-infty_in x0
proof end;

theorem :: LIMFUNC2:77
for x0 being Real
for f being PartFunc of REAL,REAL st f is_right_convergent_in x0 & lim_right (f,x0) = 0 & ( for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom f & f . g <> 0 ) ) & ex r being Real st
( 0 < r & ( for g being Real st g in (dom f) /\ ].x0,(x0 + r).[ holds
0 <= f . g ) ) holds
f ^ is_right_divergent_to+infty_in x0
proof end;

theorem :: LIMFUNC2:78
for x0 being Real
for f being PartFunc of REAL,REAL st f is_right_convergent_in x0 & lim_right (f,x0) = 0 & ( for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom f & f . g <> 0 ) ) & ex r being Real st
( 0 < r & ( for g being Real st g in (dom f) /\ ].x0,(x0 + r).[ holds
f . g <= 0 ) ) holds
f ^ is_right_divergent_to-infty_in x0
proof end;