:: GRAPH_2 semantic presentation
theorem Th1: :: GRAPH_2:1
theorem Th2: :: GRAPH_2:2
theorem Th3: :: GRAPH_2:3
Lemma69:
for m, n being Element of NAT
for F being finite set st F = { k where k is Element of NAT : ( m <= k & k <= m + n ) } holds
card F = n + 1
theorem Th4: :: GRAPH_2:4
theorem Th5: :: GRAPH_2:5
:: deftheorem Def1 defines -cut GRAPH_2:def 1 :
Lemma90:
for p being FinSequence
for m, n being Element of NAT st 1 <= m & m <= n + 1 & n <= len p holds
( (len (m,n -cut p)) + m = n + 1 & ( for i being Element of NAT st i < len (m,n -cut p) holds
(m,n -cut p) . (i + 1) = p . (m + i) ) )
theorem Th6: :: GRAPH_2:6
theorem Th7: :: GRAPH_2:7
theorem Th8: :: GRAPH_2:8
theorem Th9: :: GRAPH_2:9
theorem Th10: :: GRAPH_2:10
theorem Th11: :: GRAPH_2:11
theorem Th12: :: GRAPH_2:12
:: deftheorem Def2 defines ^' GRAPH_2:def 2 :
theorem Th13: :: GRAPH_2:13
theorem Th14: :: GRAPH_2:14
theorem Th15: :: GRAPH_2:15
theorem Th16: :: GRAPH_2:16
theorem Th17: :: GRAPH_2:17
theorem Th18: :: GRAPH_2:18
:: deftheorem Def3 defines TwoValued GRAPH_2:def 3 :
theorem Th19: :: GRAPH_2:19
then Lemma121:
<*1,2*> is TwoValued
by ;
:: deftheorem Def4 defines Alternating GRAPH_2:def 4 :
Lemma124:
<*1,2*> is Alternating
by , ;
theorem Th20: :: GRAPH_2:20
theorem Th21: :: GRAPH_2:21
theorem Th22: :: GRAPH_2:22
theorem Th23: :: GRAPH_2:23
:: deftheorem Def5 defines FinSubsequence GRAPH_2:def 5 :
theorem Th24: :: GRAPH_2:24
theorem Th25: :: GRAPH_2:25
canceled;
theorem Th26: :: GRAPH_2:26
theorem Th27: :: GRAPH_2:27
theorem Th28: :: GRAPH_2:28
theorem Th29: :: GRAPH_2:29
theorem Th30: :: GRAPH_2:30
theorem Th31: :: GRAPH_2:31
theorem Th32: :: GRAPH_2:32
:: deftheorem Def6 defines -VSet GRAPH_2:def 6 :
:: deftheorem Def7 defines is_vertex_seq_of GRAPH_2:def 7 :
theorem Th33: :: GRAPH_2:33
canceled;
theorem Th34: :: GRAPH_2:34
theorem Th35: :: GRAPH_2:35
theorem Th36: :: GRAPH_2:36
theorem Th37: :: GRAPH_2:37
:: deftheorem Def8 defines alternates_vertices_in GRAPH_2:def 8 :
theorem Th38: :: GRAPH_2:38
theorem Th39: :: GRAPH_2:39
theorem Th40: :: GRAPH_2:40
theorem Th41: :: GRAPH_2:41
Lemma213:
for D being non empty set st ( for x, y being set st x in D & y in D holds
x = y ) holds
Card D = 1
theorem Th42: :: GRAPH_2:42
:: deftheorem Def9 defines vertex-seq GRAPH_2:def 9 :
theorem Th43: :: GRAPH_2:43
theorem Th44: :: GRAPH_2:44
theorem Th45: :: GRAPH_2:45
theorem Th46: :: GRAPH_2:46
theorem Th47: :: GRAPH_2:47
Lemma231:
for G being Graph
for v being Element of the Vertices of G holds <*v*> is_vertex_seq_of {}
:: deftheorem Def10 defines simple GRAPH_2:def 10 :
theorem Th48: :: GRAPH_2:48
canceled;
theorem Th49: :: GRAPH_2:49
theorem Th50: :: GRAPH_2:50
theorem Th51: :: GRAPH_2:51
theorem Th52: :: GRAPH_2:52
theorem Th53: :: GRAPH_2:53
theorem Th54: :: GRAPH_2:54
theorem Th55: :: GRAPH_2:55
theorem Th56: :: GRAPH_2:56
:: deftheorem Def11 defines vertex-seq GRAPH_2:def 11 :
theorem Th57: :: GRAPH_2:57
theorem Th58: :: GRAPH_2:58
theorem Th59: :: GRAPH_2:59
theorem Th60: :: GRAPH_2:60
theorem Th61: :: GRAPH_2:61
theorem Th62: :: GRAPH_2:62