:: The Class of Series --- Parallel Graphs, {I}
:: by Krzysztof Retel
::
:: Received November 18, 2002
:: Copyright (c) 2002 Association of Mizar Users
theorem :: NECKLACE:1
canceled;
theorem Th2: :: NECKLACE:2
theorem Th3: :: NECKLACE:3
for
x1,
x2,
x3,
y1,
y2,
y3 being
set holds
[:{x1,x2,x3},{y1,y2,y3}:] = {[x1,y1],[x1,y2],[x1,y3],[x2,y1],[x2,y2],[x2,y3],[x3,y1],[x3,y2],[x3,y3]}
theorem Th4: :: NECKLACE:4
for
x being
set for
n being
Nat st
x in n holds
x is
Nat
theorem Th5: :: NECKLACE:5
theorem Th6: :: NECKLACE:6
theorem Th7: :: NECKLACE:7
theorem Th8: :: NECKLACE:8
theorem :: NECKLACE:9
theorem Th10: :: NECKLACE:10
theorem Th11: :: NECKLACE:11
for
a,
b,
c,
d being
set holds
not ( (
a = b implies
c = d ) & (
c = d implies
a = b ) & not
(a,b --> c,d) " = c,
d --> a,
b )
theorem :: NECKLACE:12
for
i,
j,
n being
Nat st
i < j &
j in n holds
i in n
:: deftheorem NECKLACE:def 1 :
canceled;
:: deftheorem Def2 defines embeds NECKLACE:def 2 :
theorem Th13: :: NECKLACE:13
:: deftheorem Def3 defines is_equimorphic_to NECKLACE:def 3 :
theorem :: NECKLACE:14
:: deftheorem Def4 defines symmetric NECKLACE:def 4 :
:: deftheorem Def5 defines asymmetric NECKLACE:def 5 :
theorem :: NECKLACE:15
:: deftheorem defines irreflexive NECKLACE:def 6 :
:: deftheorem Def7 defines -SuccRelStr NECKLACE:def 7 :
theorem :: NECKLACE:16
theorem Th17: :: NECKLACE:17
:: deftheorem Def8 defines SymRelStr NECKLACE:def 8 :
Lm1:
for S, R being non empty RelStr st S,R are_isomorphic holds
card the InternalRel of S = card the InternalRel of R
:: deftheorem Def9 defines ComplRelStr NECKLACE:def 9 :
theorem Th18: :: NECKLACE:18
:: deftheorem defines Necklace NECKLACE:def 10 :
theorem Th19: :: NECKLACE:19
theorem Th20: :: NECKLACE:20
theorem Th21: :: NECKLACE:21
theorem Th22: :: NECKLACE:22
theorem Th23: :: NECKLACE:23
theorem :: NECKLACE:24
theorem Th25: :: NECKLACE:25
theorem Th26: :: NECKLACE:26
theorem Th27: :: NECKLACE:27
theorem Th28: :: NECKLACE:28
theorem :: NECKLACE:29
theorem :: NECKLACE:30
theorem :: NECKLACE:31