:: FINTOPO6 semantic presentation
theorem Th1: :: FINTOPO6:1
theorem Th2: :: FINTOPO6:2
theorem Th3: :: FINTOPO6:3
:: deftheorem Def1 defines connected FINTOPO6:def 1 :
theorem Th4: :: FINTOPO6:4
theorem Th5: :: FINTOPO6:5
theorem Th6: :: FINTOPO6:6
theorem Th7: :: FINTOPO6:7
:: deftheorem Def2 defines SubSpace FINTOPO6:def 2 :
Lemma48:
for T being RelStr holds RelStr(# the carrier of T,the InternalRel of T #) is SubSpace of T
:: deftheorem Def3 defines | FINTOPO6:def 3 :
theorem Th8: :: FINTOPO6:8
theorem Th9: :: FINTOPO6:9
theorem Th10: :: FINTOPO6:10
theorem Th11: :: FINTOPO6:11
theorem Th12: :: FINTOPO6:12
theorem Th13: :: FINTOPO6:13
theorem Th14: :: FINTOPO6:14
theorem Th15: :: FINTOPO6:15
theorem Th16: :: FINTOPO6:16
theorem Th17: :: FINTOPO6:17
theorem Th18: :: FINTOPO6:18
theorem Th19: :: FINTOPO6:19
theorem Th20: :: FINTOPO6:20
theorem Th21: :: FINTOPO6:21
theorem Th22: :: FINTOPO6:22
theorem Th23: :: FINTOPO6:23
theorem Th24: :: FINTOPO6:24
theorem Th25: :: FINTOPO6:25
:: deftheorem Def4 defines is_a_component_of FINTOPO6:def 4 :
theorem Th26: :: FINTOPO6:26
theorem Th27: :: FINTOPO6:27
theorem Th28: :: FINTOPO6:28
theorem Th29: :: FINTOPO6:29
theorem Th30: :: FINTOPO6:30
theorem Th31: :: FINTOPO6:31
theorem Th32: :: FINTOPO6:32
theorem Th33: :: FINTOPO6:33
theorem Th34: :: FINTOPO6:34
theorem Th35: :: FINTOPO6:35
theorem Th36: :: FINTOPO6:36
theorem Th37: :: FINTOPO6:37
theorem Th38: :: FINTOPO6:38
theorem Th39: :: FINTOPO6:39
theorem Th40: :: FINTOPO6:40
theorem Th41: :: FINTOPO6:41
theorem Th42: :: FINTOPO6:42
:: deftheorem Def5 defines is_a_component_of FINTOPO6:def 5 :
theorem Th43: :: FINTOPO6:43
:: deftheorem Def6 defines continuous FINTOPO6:def 6 :
Lemma143:
for FT being non empty RelStr
for x being Element of FT holds <*x*> is continuous
theorem Th44: :: FINTOPO6:44
theorem Th45: :: FINTOPO6:45
:: deftheorem Def7 defines arcwise_connected FINTOPO6:def 7 :
Lemma150:
for FT being non empty RelStr holds {} FT is arcwise_connected
Lemma151:
for FT being non empty RelStr
for x being Element of FT holds {x} is arcwise_connected
theorem Th46: :: FINTOPO6:46
theorem Th47: :: FINTOPO6:47
theorem Th48: :: FINTOPO6:48
:: deftheorem Def8 defines is_minimum_path_in FINTOPO6:def 8 :
theorem Th49: :: FINTOPO6:49
Lemma166:
for FT being non empty RelStr
for f being FinSequence of FT
for A being Subset of FT
for x1, x2 being Element of FT st f is continuous & rng f c= A & f . 1 = x1 & f . (len f) = x2 holds
ex g being FinSequence of FT st
( g is continuous & rng g c= A & g . 1 = x1 & g . (len g) = x2 & ( for h being FinSequence of FT st h is continuous & rng h c= A & h . 1 = x1 & h . (len h) = x2 holds
len g <= len h ) )
theorem Th50: :: FINTOPO6:50
theorem Th51: :: FINTOPO6:51
theorem Th52: :: FINTOPO6:52
theorem Th53: :: FINTOPO6:53
theorem Th54: :: FINTOPO6:54
theorem Th55: :: FINTOPO6:55
:: deftheorem Def9 defines inv_continuous FINTOPO6:def 9 :
theorem Th56: :: FINTOPO6:56
theorem Th57: :: FINTOPO6:57
theorem Th58: :: FINTOPO6:58