:: GRAPH_1 semantic presentation
set DomEx = {1,2};
reconsider empty1 = {} as Function of {} ,{1,2} by FUNCT_2:55, RELAT_1:60;
:: deftheorem Def1 defines Graph-like GRAPH_1:def 1 :
Lemma27:
for G being Graph holds
( dom the Source of G = the Edges of G & dom the Target of G = the Edges of G )
Lemma28:
for G being Graph
for x being Element of the Vertices of G holds x in the Vertices of G
:: deftheorem Def2 defines \/ GRAPH_1:def 2 :
:: deftheorem Def3 defines is_sum_of GRAPH_1:def 3 :
:: deftheorem Def4 defines oriented GRAPH_1:def 4 :
:: deftheorem Def5 defines non-multi GRAPH_1:def 5 :
:: deftheorem Def6 defines simple GRAPH_1:def 6 :
:: deftheorem Def7 defines connected GRAPH_1:def 7 :
:: deftheorem Def8 defines finite GRAPH_1:def 8 :
:: deftheorem Def9 defines joins GRAPH_1:def 9 :
:: deftheorem Def10 defines are_incydent GRAPH_1:def 10 :
:: deftheorem Def11 defines Chain GRAPH_1:def 11 :
Lemma83:
for G being Graph holds {} is Chain of G
:: deftheorem Def12 defines oriented GRAPH_1:def 12 :
Lemma86:
for G being Graph holds {} is oriented Chain of G
:: deftheorem Def13 defines one-to-one GRAPH_1:def 13 :
:: deftheorem Def14 GRAPH_1:def 14 :
canceled;
:: deftheorem Def15 defines cyclic GRAPH_1:def 15 :
Lemma90:
for G being Graph holds {} is Cycle of G
Lemma92:
for G being Graph
for v being set st v in the Edges of G holds
( the Source of G . v in the Vertices of G & the Target of G . v in the Vertices of G )
:: deftheorem Def16 GRAPH_1:def 16 :
canceled;
:: deftheorem Def17 defines Subgraph GRAPH_1:def 17 :
:: deftheorem Def18 defines VerticesCount GRAPH_1:def 18 :
:: deftheorem Def19 defines EdgesCount GRAPH_1:def 19 :
:: deftheorem Def20 defines EdgesIn GRAPH_1:def 20 :
:: deftheorem Def21 defines EdgesOut GRAPH_1:def 21 :
:: deftheorem Def22 defines Degree GRAPH_1:def 22 :
Lemma100:
for n being Element of NAT
for G being Graph
for p being Chain of G holds p | (Seg n) is Chain of G
Lemma103:
for G being Graph
for H1, H2 being strict Subgraph of G st the Vertices of H1 = the Vertices of H2 & the Edges of H1 = the Edges of H2 holds
H1 = H2
:: deftheorem Def23 defines c= GRAPH_1:def 23 :
Lemma107:
for G being Graph
for H being Subgraph of G holds
( the Source of H in PFuncs the Edges of G,the Vertices of G & the Target of H in PFuncs the Edges of G,the Vertices of G )
:: deftheorem Def24 defines bool GRAPH_1:def 24 :
theorem Th1: :: GRAPH_1:1
theorem Th2: :: GRAPH_1:2
theorem Th3: :: GRAPH_1:3
theorem Th4: :: GRAPH_1:4
theorem Th5: :: GRAPH_1:5
theorem Th6: :: GRAPH_1:6
theorem Th7: :: GRAPH_1:7
theorem Th8: :: GRAPH_1:8
theorem Th9: :: GRAPH_1:9
theorem Th10: :: GRAPH_1:10
theorem Th11: :: GRAPH_1:11
theorem Th12: :: GRAPH_1:12
theorem Th13: :: GRAPH_1:13
theorem Th14: :: GRAPH_1:14
theorem Th15: :: GRAPH_1:15
theorem Th16: :: GRAPH_1:16
theorem Th17: :: GRAPH_1:17
for
G1,
G2,
G3 being
Graph st
G1 c= G2 &
G2 c= G3 holds
G1 c= G3
theorem Th18: :: GRAPH_1:18
theorem Th19: :: GRAPH_1:19
theorem Th20: :: GRAPH_1:20
theorem Th21: :: GRAPH_1:21
theorem Th22: :: GRAPH_1:22
for
G1,
G,
G2 being
Graph st
G1 c= G &
G2 c= G holds
G1 \/ G2 c= G
theorem Th23: :: GRAPH_1:23
theorem Th24: :: GRAPH_1:24
theorem Th25: :: GRAPH_1:25
canceled;
theorem Th26: :: GRAPH_1:26
canceled;
theorem Th27: :: GRAPH_1:27
theorem Th28: :: GRAPH_1:28
theorem Th29: :: GRAPH_1:29
theorem Th30: :: GRAPH_1:30
theorem Th31: :: GRAPH_1:31
theorem Th32: :: GRAPH_1:32
theorem Th33: :: GRAPH_1:33
canceled;
theorem Th34: :: GRAPH_1:34
theorem Th35: :: GRAPH_1:35
theorem Th36: :: GRAPH_1:36
theorem Th37: :: GRAPH_1:37