:: FRECHET semantic presentation
Lemma1:
for b1 being Nat holds
not ( b1 <> 0 & not 1 / b1 > 0 )
Lemma2:
for b1 being Real holds
not ( b1 > 0 & ( for b2 being Nat holds
not ( 1 / b2 < b1 & b2 > 0 ) ) )
theorem Th1: :: FRECHET:1
theorem Th2: :: FRECHET:2
theorem Th3: :: FRECHET:3
Lemma5:
for b1 being TopStruct
for b2 being Subset of b1 holds
( b2 is open iff ([#] b1) \ b2 is closed )
theorem Th4: :: FRECHET:4
theorem Th5: :: FRECHET:5
theorem Th6: :: FRECHET:6
theorem Th7: :: FRECHET:7
theorem Th8: :: FRECHET:8
for b
1 being
Subset of
R^1 holds
( b
1 is
open iff for b
2 being
Real holds
not ( b
2 in b
1 & ( for b
3 being
Real holds
not ( b
3 > 0 &
].(b2 - b3),(b2 + b3).[ c= b
1 ) ) ) )
theorem Th9: :: FRECHET:9
theorem Th10: :: FRECHET:10
theorem Th11: :: FRECHET:11
theorem Th12: :: FRECHET:12
theorem Th13: :: FRECHET:13
for b
1, b
2 being
set holds
( b
2 c= b
1 implies
(id b1) .: b
2 = b
2 )
theorem Th14: :: FRECHET:14
canceled;
theorem Th15: :: FRECHET:15
theorem Th16: :: FRECHET:16
theorem Th17: :: FRECHET:17
for b
1, b
2, b
3, b
4 being
set holds
( b
3 c= b
1 implies
((id b1) +* (b2 --> b4)) " (b3 \ {b4}) = (b3 \ b2) \ {b4} )
theorem Th18: :: FRECHET:18
for b
1, b
2, b
3 being
set holds
( not b
3 in b
1 implies
((id b1) +* (b2 --> b3)) " {b3} = b
2 )
theorem Th19: :: FRECHET:19
for b
1, b
2, b
3, b
4 being
set holds
( b
3 c= b
1 & not b
4 in b
1 implies
((id b1) +* (b2 --> b4)) " (b3 \/ {b4}) = b
3 \/ b
2 )
theorem Th20: :: FRECHET:20
for b
1, b
2, b
3, b
4 being
set holds
( b
3 c= b
1 & not b
4 in b
1 implies
((id b1) +* (b2 --> b4)) " (b3 \ {b4}) = b
3 \ b
2 )
Lemma21:
for b1, b2, b3, b4 being set holds
( not b4 in b1 implies ((id b1) +* (b2 --> b4)) " ((b1 \ b3) \ {b4}) = (b1 \ b3) \ b2 )
by Th20;
:: deftheorem Def1 defines first-countable FRECHET:def 1 :
theorem Th21: :: FRECHET:21
theorem Th22: :: FRECHET:22
:: deftheorem Def2 defines is_convergent_to FRECHET:def 2 :
theorem Th23: :: FRECHET:23
:: deftheorem Def3 defines convergent FRECHET:def 3 :
:: deftheorem Def4 defines Lim FRECHET:def 4 :
:: deftheorem Def5 defines Frechet FRECHET:def 5 :
:: deftheorem Def6 defines sequential FRECHET:def 6 :
theorem Th24: :: FRECHET:24
theorem Th25: :: FRECHET:25
canceled;
theorem Th26: :: FRECHET:26
theorem Th27: :: FRECHET:27
theorem Th28: :: FRECHET:28
:: deftheorem Def7 defines REAL? FRECHET:def 7 :
Lemma34:
{REAL } c= the carrier of REAL?
theorem Th29: :: FRECHET:29
canceled;
theorem Th30: :: FRECHET:30
theorem Th31: :: FRECHET:31
theorem Th32: :: FRECHET:32
theorem Th33: :: FRECHET:33
theorem Th34: :: FRECHET:34
theorem Th35: :: FRECHET:35
theorem Th36: :: FRECHET:36
theorem Th37: :: FRECHET:37
theorem Th38: :: FRECHET:38
canceled;
theorem Th39: :: FRECHET:39
theorem Th40: :: FRECHET:40
for b
1 being
Real holds
not ( b
1 > 0 & ( for b
2 being
Nat holds
not ( 1
/ b
2 < b
1 & b
2 > 0 ) ) )
by Lemma2;