:: LIMFUNC3 semantic presentation
Lemma1:
for b1, b2, b3 being set holds
( b1 c= b2 \ b3 implies b1 c= b2 )
by XBOOLE_1:1;
Lemma2:
for b1, b2, b3 being real number holds
( 0 < b1 & b2 <= b3 implies ( b2 - b1 < b3 & b2 < b3 + b1 ) )
Lemma3:
for b1 being Real_Sequence
for b2, b3 being PartFunc of REAL , REAL
for b4 being set holds
( rng b1 c= (dom (b2 (#) b3)) \ b4 implies ( rng b1 c= dom (b2 (#) b3) & dom (b2 (#) b3) = (dom b2) /\ (dom b3) & rng b1 c= dom b2 & rng b1 c= dom b3 & rng b1 c= (dom b2) \ b4 & rng b1 c= (dom b3) \ b4 ) )
Lemma4:
for b1 being Real
for b2 being Nat holds
( b1 - (1 / (b2 + 1)) < b1 & b1 < b1 + (1 / (b2 + 1)) )
Lemma5:
for b1 being Nat holds
0 < 1 / (b1 + 1)
by XREAL_1:141;
Lemma6:
for b1 being Real_Sequence
for b2, b3 being PartFunc of REAL , REAL
for b4 being set holds
( rng b1 c= (dom (b2 + b3)) \ b4 implies ( rng b1 c= dom (b2 + b3) & dom (b2 + b3) = (dom b2) /\ (dom b3) & rng b1 c= dom b2 & rng b1 c= dom b3 & rng b1 c= (dom b2) \ b4 & rng b1 c= (dom b3) \ b4 ) )
theorem Th1: :: LIMFUNC3:1
theorem Th2: :: LIMFUNC3:2
theorem Th3: :: LIMFUNC3:3
theorem Th4: :: LIMFUNC3:4
theorem Th5: :: LIMFUNC3:5
theorem Th6: :: LIMFUNC3:6
theorem Th7: :: LIMFUNC3:7
theorem Th8: :: LIMFUNC3:8
for b
1 being
Realfor b
2 being
PartFunc of
REAL ,
REAL holds
( ( for b
3, b
4 being
Real holds
not ( b
3 < b
1 & b
1 < b
4 & ( for b
5, b
6 being
Real holds
not ( b
3 < b
5 & b
5 < b
1 & b
5 in dom b
2 & b
6 < b
4 & b
1 < b
6 & b
6 in dom b
2 ) ) ) ) iff ( ( for b
3 being
Real holds
not ( b
3 < b
1 & ( for b
4 being
Real holds
not ( b
3 < b
4 & b
4 < b
1 & b
4 in dom b
2 ) ) ) ) & ( for b
3 being
Real holds
not ( b
1 < b
3 & ( for b
4 being
Real holds
not ( b
4 < b
3 & b
1 < b
4 & b
4 in dom b
2 ) ) ) ) ) )
:: deftheorem Def1 defines is_convergent_in LIMFUNC3:def 1 :
:: deftheorem Def2 defines is_divergent_to+infty_in LIMFUNC3:def 2 :
:: deftheorem Def3 defines is_divergent_to-infty_in LIMFUNC3:def 3 :
theorem Th9: :: LIMFUNC3:9
canceled;
theorem Th10: :: LIMFUNC3:10
canceled;
theorem Th11: :: LIMFUNC3:11
canceled;
theorem Th12: :: LIMFUNC3:12
theorem Th13: :: LIMFUNC3:13
theorem Th14: :: LIMFUNC3:14
theorem Th15: :: LIMFUNC3:15
theorem Th16: :: LIMFUNC3:16
theorem Th17: :: LIMFUNC3:17
theorem Th18: :: LIMFUNC3:18
theorem Th19: :: LIMFUNC3:19
theorem Th20: :: LIMFUNC3:20
theorem Th21: :: LIMFUNC3:21
theorem Th22: :: LIMFUNC3:22
theorem Th23: :: LIMFUNC3:23
theorem Th24: :: LIMFUNC3:24
theorem Th25: :: LIMFUNC3:25
theorem Th26: :: LIMFUNC3:26
theorem Th27: :: LIMFUNC3:27
for b
1 being
Realfor b
2, b
3 being
PartFunc of
REAL ,
REAL holds
( b
2 is_divergent_to+infty_in b
1 & ( for b
4, b
5 being
Real holds
not ( b
4 < b
1 & b
1 < b
5 & ( for b
6, b
7 being
Real holds
not ( b
4 < b
6 & b
6 < b
1 & b
6 in dom b
3 & b
7 < b
5 & b
1 < b
7 & b
7 in dom b
3 ) ) ) ) & ex b
4 being
Real st
( 0
< b
4 &
(dom b3) /\ (].(b1 - b4),b1.[ \/ ].b1,(b1 + b4).[) c= (dom b2) /\ (].(b1 - b4),b1.[ \/ ].b1,(b1 + b4).[) & ( for b
5 being
Real holds
( b
5 in (dom b3) /\ (].(b1 - b4),b1.[ \/ ].b1,(b1 + b4).[) implies b
2 . b
5 <= b
3 . b
5 ) ) ) implies b
3 is_divergent_to+infty_in b
1 )
theorem Th28: :: LIMFUNC3:28
for b
1 being
Realfor b
2, b
3 being
PartFunc of
REAL ,
REAL holds
( b
2 is_divergent_to-infty_in b
1 & ( for b
4, b
5 being
Real holds
not ( b
4 < b
1 & b
1 < b
5 & ( for b
6, b
7 being
Real holds
not ( b
4 < b
6 & b
6 < b
1 & b
6 in dom b
3 & b
7 < b
5 & b
1 < b
7 & b
7 in dom b
3 ) ) ) ) & ex b
4 being
Real st
( 0
< b
4 &
(dom b3) /\ (].(b1 - b4),b1.[ \/ ].b1,(b1 + b4).[) c= (dom b2) /\ (].(b1 - b4),b1.[ \/ ].b1,(b1 + b4).[) & ( for b
5 being
Real holds
( b
5 in (dom b3) /\ (].(b1 - b4),b1.[ \/ ].b1,(b1 + b4).[) implies b
3 . b
5 <= b
2 . b
5 ) ) ) implies b
3 is_divergent_to-infty_in b
1 )
theorem Th29: :: LIMFUNC3:29
theorem Th30: :: LIMFUNC3:30
:: deftheorem Def4 defines lim LIMFUNC3:def 4 :
theorem Th31: :: LIMFUNC3:31
canceled;
theorem Th32: :: LIMFUNC3:32
theorem Th33: :: LIMFUNC3:33
theorem Th34: :: LIMFUNC3:34
theorem Th35: :: LIMFUNC3:35
theorem Th36: :: LIMFUNC3:36
theorem Th37: :: LIMFUNC3:37
theorem Th38: :: LIMFUNC3:38
theorem Th39: :: LIMFUNC3:39
theorem Th40: :: LIMFUNC3:40
theorem Th41: :: LIMFUNC3:41
theorem Th42: :: LIMFUNC3:42
theorem Th43: :: LIMFUNC3:43
theorem Th44: :: LIMFUNC3:44
theorem Th45: :: LIMFUNC3:45
for b
1 being
Realfor b
2, b
3, b
4 being
PartFunc of
REAL ,
REAL holds
( b
2 is_convergent_in b
1 & b
3 is_convergent_in b
1 &
lim b
2,b
1 = lim b
3,b
1 & ( for b
5, b
6 being
Real holds
not ( b
5 < b
1 & b
1 < b
6 & ( for b
7, b
8 being
Real holds
not ( b
5 < b
7 & b
7 < b
1 & b
7 in dom b
4 & b
8 < b
6 & b
1 < b
8 & b
8 in dom b
4 ) ) ) ) & ex b
5 being
Real st
( 0
< b
5 & ( for b
6 being
Real holds
( b
6 in (dom b4) /\ (].(b1 - b5),b1.[ \/ ].b1,(b1 + b5).[) implies ( b
2 . b
6 <= b
4 . b
6 & b
4 . b
6 <= b
3 . b
6 ) ) ) & ( (
(dom b2) /\ (].(b1 - b5),b1.[ \/ ].b1,(b1 + b5).[) c= (dom b3) /\ (].(b1 - b5),b1.[ \/ ].b1,(b1 + b5).[) &
(dom b4) /\ (].(b1 - b5),b1.[ \/ ].b1,(b1 + b5).[) c= (dom b2) /\ (].(b1 - b5),b1.[ \/ ].b1,(b1 + b5).[) ) or (
(dom b3) /\ (].(b1 - b5),b1.[ \/ ].b1,(b1 + b5).[) c= (dom b2) /\ (].(b1 - b5),b1.[ \/ ].b1,(b1 + b5).[) &
(dom b4) /\ (].(b1 - b5),b1.[ \/ ].b1,(b1 + b5).[) c= (dom b3) /\ (].(b1 - b5),b1.[ \/ ].b1,(b1 + b5).[) ) ) ) implies ( b
4 is_convergent_in b
1 &
lim b
4,b
1 = lim b
2,b
1 ) )
theorem Th46: :: LIMFUNC3:46
for b
1 being
Realfor b
2, b
3, b
4 being
PartFunc of
REAL ,
REAL holds
( b
2 is_convergent_in b
1 & b
3 is_convergent_in b
1 &
lim b
2,b
1 = lim b
3,b
1 & ex b
5 being
Real st
( 0
< b
5 &
].(b1 - b5),b1.[ \/ ].b1,(b1 + b5).[ c= ((dom b2) /\ (dom b3)) /\ (dom b4) & ( for b
6 being
Real holds
( b
6 in ].(b1 - b5),b1.[ \/ ].b1,(b1 + b5).[ implies ( b
2 . b
6 <= b
4 . b
6 & b
4 . b
6 <= b
3 . b
6 ) ) ) ) implies ( b
4 is_convergent_in b
1 &
lim b
4,b
1 = lim b
2,b
1 ) )
theorem Th47: :: LIMFUNC3:47
for b
1 being
Realfor b
2, b
3 being
PartFunc of
REAL ,
REAL holds
( b
2 is_convergent_in b
1 & b
3 is_convergent_in b
1 & ex b
4 being
Real st
( 0
< b
4 & ( (
(dom b2) /\ (].(b1 - b4),b1.[ \/ ].b1,(b1 + b4).[) c= (dom b3) /\ (].(b1 - b4),b1.[ \/ ].b1,(b1 + b4).[) & ( for b
5 being
Real holds
( b
5 in (dom b2) /\ (].(b1 - b4),b1.[ \/ ].b1,(b1 + b4).[) implies b
2 . b
5 <= b
3 . b
5 ) ) ) or (
(dom b3) /\ (].(b1 - b4),b1.[ \/ ].b1,(b1 + b4).[) c= (dom b2) /\ (].(b1 - b4),b1.[ \/ ].b1,(b1 + b4).[) & ( for b
5 being
Real holds
( b
5 in (dom b3) /\ (].(b1 - b4),b1.[ \/ ].b1,(b1 + b4).[) implies b
2 . b
5 <= b
3 . b
5 ) ) ) ) ) implies
lim b
2,b
1 <= lim b
3,b
1 )
theorem Th48: :: LIMFUNC3:48
theorem Th49: :: LIMFUNC3:49
theorem Th50: :: LIMFUNC3:50
theorem Th51: :: LIMFUNC3:51
theorem Th52: :: LIMFUNC3:52