:: FRECHET2 semantic presentation
Lemma24:
for T being non empty TopSpace st ( for p being Point of T holds Cl {p} = {p} ) holds
T is_T1
Lemma28:
for T being non empty TopSpace st not T is_T1 holds
ex x1, x2 being Point of T st
( x1 <> x2 & x2 in Cl {x1} )
Lemma33:
for T being non empty TopSpace st not T is_T1 holds
ex x1, x2 being Point of T ex S being sequence of T st
( S = NAT --> x1 & x1 <> x2 & S is_convergent_to x2 )
Lemma36:
for T being non empty TopSpace st T is_T2 holds
T is_T1
Lemma37:
for T being non empty 1-sorted
for S being sequence of T
for f being Function of NAT , NAT holds S * f is sequence of T
;
theorem Th1: :: FRECHET2:1
Lemma42:
id NAT is Real_Sequence
Lemma45:
for RS being Real_Sequence st RS = id NAT holds
RS is natural-yielding
Lemma47:
for RS being Real_Sequence st RS = id NAT holds
RS is increasing
theorem Th2: :: FRECHET2:2
Lemma48:
for T being non empty 1-sorted
for S being sequence of T ex NS being increasing Seq_of_Nat st S = S * NS
theorem Th3: :: FRECHET2:3
theorem Th4: :: FRECHET2:4
Lemma54:
for T being non empty 1-sorted
for S being sequence of T
for NS being increasing Seq_of_Nat holds S * NS is subsequence of S
theorem Th5: :: FRECHET2:5
theorem Th6: :: FRECHET2:6
theorem Th7: :: FRECHET2:7
Lemma94:
for T being non empty TopSpace st T is first-countable holds
for x being Point of T ex B being Basis of x ex S being Function st
( dom S = NAT & rng S = B & ( for n, m being Element of NAT st m >= n holds
S . m c= S . n ) )
theorem Th8: :: FRECHET2:8
theorem Th9: :: FRECHET2:9
theorem Th10: :: FRECHET2:10
theorem Th11: :: FRECHET2:11
theorem Th12: :: FRECHET2:12
theorem Th13: :: FRECHET2:13
theorem Th14: :: FRECHET2:14
theorem Th15: :: FRECHET2:15
:: deftheorem Def1 FRECHET2:def 1 :
canceled;
:: deftheorem Def2 defines Cl_Seq FRECHET2:def 2 :
theorem Th16: :: FRECHET2:16
theorem Th17: :: FRECHET2:17
theorem Th18: :: FRECHET2:18
theorem Th19: :: FRECHET2:19
theorem Th20: :: FRECHET2:20
theorem Th21: :: FRECHET2:21
theorem Th22: :: FRECHET2:22
theorem Th23: :: FRECHET2:23
theorem Th24: :: FRECHET2:24
theorem Th25: :: FRECHET2:25
theorem Th26: :: FRECHET2:26
:: deftheorem Def3 defines lim FRECHET2:def 3 :
theorem Th27: :: FRECHET2:27
theorem Th28: :: FRECHET2:28
theorem Th29: :: FRECHET2:29
theorem Th30: :: FRECHET2:30
theorem Th31: :: FRECHET2:31
theorem Th32: :: FRECHET2:32
theorem Th33: :: FRECHET2:33
:: deftheorem Def4 defines is_a_cluster_point_of FRECHET2:def 4 :
theorem Th34: :: FRECHET2:34
theorem Th35: :: FRECHET2:35
theorem Th36: :: FRECHET2:36
theorem Th37: :: FRECHET2:37
theorem Th38: :: FRECHET2:38
Lemma166:
for f being Function st not dom f is finite & f is one-to-one holds
not rng f is finite
theorem Th39: :: FRECHET2:39
theorem Th40: :: FRECHET2:40
theorem Th41: :: FRECHET2:41
theorem Th42: :: FRECHET2:42
theorem Th43: :: FRECHET2:43
theorem Th44: :: FRECHET2:44
theorem Th45: :: FRECHET2:45
theorem Th46: :: FRECHET2:46
theorem Th47: :: FRECHET2:47
theorem Th48: :: FRECHET2:48
theorem Th49: :: FRECHET2:49
theorem Th50: :: FRECHET2:50