:: GLIB_001 semantic presentation
theorem Th1: :: GLIB_001:1
Lemma30:
for x, y, z being real number st 0 < z & x * z <= y * z holds
x <= y
by XREAL_1:70;
Lemma32:
for F being finite Function holds card (dom F) = card F
by PRE_CIRC:21;
theorem Th2: :: GLIB_001:2
theorem Th3: :: GLIB_001:3
theorem Th4: :: GLIB_001:4
theorem Th5: :: GLIB_001:5
theorem Th6: :: GLIB_001:6
:: deftheorem Def1 defines VertexSeq GLIB_001:def 1 :
:: deftheorem Def2 defines EdgeSeq GLIB_001:def 2 :
:: deftheorem Def3 defines Walk GLIB_001:def 3 :
:: deftheorem Def4 defines .walkOf GLIB_001:def 4 :
definition
let G be
_Graph;
let x be
set ,
y be
set ,
e be
set ;
func c1 .walkOf c2,
c4,
c3 -> Walk of
a1 equals :
Def5:
:: GLIB_001:def 5
<*x,e,y*> if e Joins x,
y,
G otherwise G .walkOf (choose (the_Vertices_of G));
coherence
( ( e Joins x,y,G implies <*x,e,y*> is Walk of G ) & ( not e Joins x,y,G implies G .walkOf (choose (the_Vertices_of G)) is Walk of G ) )
consistency
for b1 being Walk of G holds verum
;
end;
:: deftheorem Def5 defines .walkOf GLIB_001:def 5 :
for
G being
_Graph for
x,
y,
e being
set holds
( (
e Joins x,
y,
G implies
G .walkOf x,
e,
y = <*x,e,y*> ) & ( not
e Joins x,
y,
G implies
G .walkOf x,
e,
y = G .walkOf (choose (the_Vertices_of G)) ) );
:: deftheorem Def6 defines .first() GLIB_001:def 6 :
:: deftheorem Def7 defines .last() GLIB_001:def 7 :
:: deftheorem Def8 defines .vertexAt GLIB_001:def 8 :
:: deftheorem Def9 defines .reverse() GLIB_001:def 9 :
:: deftheorem Def10 defines .append GLIB_001:def 10 :
:: deftheorem Def11 defines .cut GLIB_001:def 11 :
:: deftheorem Def12 defines .remove GLIB_001:def 12 :
:: deftheorem Def13 defines .addEdge GLIB_001:def 13 :
:: deftheorem Def14 defines .vertexSeq() GLIB_001:def 14 :
:: deftheorem Def15 defines .edgeSeq() GLIB_001:def 15 :
:: deftheorem Def16 defines .vertices() GLIB_001:def 16 :
:: deftheorem Def17 defines .edges() GLIB_001:def 17 :
:: deftheorem Def18 defines .length() GLIB_001:def 18 :
:: deftheorem Def19 defines .find GLIB_001:def 19 :
:: deftheorem Def20 defines .find GLIB_001:def 20 :
:: deftheorem Def21 defines .rfind GLIB_001:def 21 :
:: deftheorem Def22 defines .rfind GLIB_001:def 22 :
:: deftheorem Def23 defines is_Walk_from GLIB_001:def 23 :
:: deftheorem Def24 defines closed GLIB_001:def 24 :
:: deftheorem Def25 defines directed GLIB_001:def 25 :
:: deftheorem Def26 defines trivial GLIB_001:def 26 :
:: deftheorem Def27 defines Trail-like GLIB_001:def 27 :
:: deftheorem Def28 defines Path-like GLIB_001:def 28 :
:: deftheorem Def29 defines vertex-distinct GLIB_001:def 29 :
:: deftheorem Def30 defines Circuit-like GLIB_001:def 30 :
:: deftheorem Def31 defines Cycle-like GLIB_001:def 31 :
Lemma142:
for G being _Graph
for W being Walk of G
for n being odd Element of NAT st n <= len W holds
W . n in the_Vertices_of G
Lemma143:
for G being _Graph
for W being Walk of G
for n being even Element of NAT st n in dom W holds
ex naa1 being odd Element of NAT st
( naa1 = n - 1 & n - 1 in dom W & n + 1 in dom W & W . n Joins W . naa1,W . (n + 1),G )
Lemma145:
for G being _Graph
for W being Walk of G
for n being odd Element of NAT st n < len W holds
( n in dom W & n + 1 in dom W & n + 2 in dom W )
Lemma146:
for G being _Graph
for v being Vertex of G holds
( G .walkOf v is closed & G .walkOf v is directed & G .walkOf v is trivial & G .walkOf v is Trail-like & G .walkOf v is Path-like )
Lemma147:
for G being _Graph
for x, e, y being set st e Joins x,y,G holds
len (G .walkOf x,e,y) = 3
Lemma148:
for G being _Graph
for x, e, y being set st e Joins x,y,G holds
( (G .walkOf x,e,y) .first() = x & (G .walkOf x,e,y) .last() = y & G .walkOf x,e,y is_Walk_from x,y )
Lemma149:
for G being _Graph
for W being Walk of G holds
( len W = len (W .reverse() ) & dom W = dom (W .reverse() ) & rng W = rng (W .reverse() ) )
by FINSEQ_5:def 3, FINSEQ_5:60;
Lemma150:
for G being _Graph
for W being Walk of G holds
( W .first() = (W .reverse() ) .last() & W .last() = (W .reverse() ) .first() )
Lemma151:
for G being _Graph
for W being Walk of G
for n being Element of NAT st n in dom (W .reverse() ) holds
( (W .reverse() ) . n = W . (((len W) - n) + 1) & ((len W) - n) + 1 in dom W )
Lemma152:
for G being _Graph
for W being Walk of G holds (W .reverse() ) .reverse() = W
by FINSEQ_6:29;
Lemma153:
for G being _Graph
for W1, W2 being Walk of G st W1 .last() = W2 .first() holds
(len (W1 .append W2)) + 1 = (len W1) + (len W2)
Lemma154:
for G being _Graph
for W1, W2 being Walk of G st W1 .last() = W2 .first() holds
( len W1 <= len (W1 .append W2) & len W2 <= len (W1 .append W2) )
Lemma155:
for G being _Graph
for W1, W2 being Walk of G st W1 .last() = W2 .first() holds
( (W1 .append W2) .first() = W1 .first() & (W1 .append W2) .last() = W2 .last() & W1 .append W2 is_Walk_from W1 .first() ,W2 .last() )
Lemma156:
for G being _Graph
for W1, W2 being Walk of G
for n being Element of NAT st n in dom W1 holds
( (W1 .append W2) . n = W1 . n & n in dom (W1 .append W2) )
Lemma157:
for G being _Graph
for W1, W2 being Walk of G st W1 .last() = W2 .first() holds
for n being Element of NAT st n < len W2 holds
( (W1 .append W2) . ((len W1) + n) = W2 . (n + 1) & (len W1) + n in dom (W1 .append W2) )
Lemma158:
for G being _Graph
for W1, W2 being Walk of G
for n being Element of NAT holds
( not n in dom (W1 .append W2) or n in dom W1 or ex k being Element of NAT st
( k < len W2 & n = (len W1) + k ) )
Lemma160:
for G being _Graph
for W being Walk of G
for m, n being odd Element of NAT st m <= n & n <= len W holds
( (len (W .cut m,n)) + m = n + 1 & ( for i being Element of NAT st i < len (W .cut m,n) holds
( (W .cut m,n) . (i + 1) = W . (m + i) & m + i in dom W ) ) )
Lemma161:
for G being _Graph
for W being Walk of G
for m, n being odd Element of NAT st m <= n & n <= len W holds
( (W .cut m,n) .first() = W . m & (W .cut m,n) .last() = W . n & W .cut m,n is_Walk_from W . m,W . n )
Lemma163:
for G being _Graph
for W being Walk of G
for m, n, o being odd Element of NAT st m <= n & n <= o & o <= len W holds
(W .cut m,n) .append (W .cut n,o) = W .cut m,o
Lemma167:
for G being _Graph
for W being Walk of G holds W .cut 1,(len W) = W
Lemma168:
for G being _Graph
for W being Walk of G
for n being odd Element of NAT st n <= len W holds
W .cut n,n = <*(W .vertexAt n)*>
Lemma169:
for G being _Graph
for W being Walk of G
for m, n being Element of NAT st not m is even & m <= n holds
(W .cut 1,n) .cut 1,m = W .cut 1,m
Lemma170:
for G being _Graph
for W1, W2 being Walk of G
for m, n being odd Element of NAT st m <= n & n <= len W1 & W1 .last() = W2 .first() holds
(W1 .append W2) .cut m,n = W1 .cut m,n
Lemma171:
for G being _Graph
for W being Walk of G
for m being odd Element of NAT st m <= len W holds
len (W .cut 1,m) = m
Lemma172:
for G being _Graph
for W being Walk of G
for m being odd Element of NAT
for x being Element of NAT st x in dom (W .cut 1,m) & m <= len W holds
(W .cut 1,m) . x = W . x
Lemma173:
for G being _Graph
for W being Walk of G
for m, n being odd Element of NAT st m <= n & n <= len W & W . m = W . n holds
(len (W .remove m,n)) + n = (len W) + m
Lemma174:
for G being _Graph
for W being Walk of G
for m, n being Element of NAT
for x, y being set st W is_Walk_from x,y holds
W .remove m,n is_Walk_from x,y
Lemma177:
for G being _Graph
for W being Walk of G
for m, n being Element of NAT holds len (W .remove m,n) <= len W
Lemma178:
for G being _Graph
for W being Walk of G
for m being Element of NAT holds W .remove m,m = W
Lemma179:
for G being _Graph
for W being Walk of G
for m, n being odd Element of NAT st m <= n & n <= len W & W . m = W . n holds
(W .cut 1,m) .last() = (W .cut n,(len W)) .first()
Lemma180:
for G being _Graph
for W being Walk of G
for x, y being set
for m, n being odd Element of NAT st m <= n & n <= len W & W . m = W . n holds
for x being Element of NAT st x in Seg m holds
(W .remove m,n) . x = W . x
Lemma181:
for G being _Graph
for W being Walk of G
for m, n being odd Element of NAT st m <= n & n <= len W & W . m = W . n holds
for x being Element of NAT st m <= x & x <= len (W .remove m,n) holds
( (W .remove m,n) . x = W . ((x - m) + n) & (x - m) + n is Element of NAT & (x - m) + n <= len W )
Lemma183:
for G being _Graph
for W being Walk of G
for m, n being odd Element of NAT st m <= n & n <= len W & W . m = W . n holds
len (W .remove m,n) = ((len W) + m) - n
Lemma184:
for G being _Graph
for W being Walk of G
for m being Element of NAT st W .first() = W . m holds
W .remove 1,m = W .cut m,(len W)
Lemma185:
for G being _Graph
for W being Walk of G
for m, n being Element of NAT holds
( (W .remove m,n) .first() = W .first() & (W .remove m,n) .last() = W .last() )
Lemma186:
for G being _Graph
for W being Walk of G
for m, n being odd Element of NAT
for x being Element of NAT st m <= n & n <= len W & W . m = W . n & x in dom (W .remove m,n) & not x in Seg m holds
( m <= x & x <= len (W .remove m,n) )
Lemma187:
for G being _Graph
for W being Walk of G
for e, x being set st e Joins W .last() ,x,G holds
W .addEdge e = W ^ <*e,x*>
Lemma189:
for G being _Graph
for W being Walk of G
for e, x being set st e Joins W .last() ,x,G holds
( (W .addEdge e) .first() = W .first() & (W .addEdge e) .last() = x & W .addEdge e is_Walk_from W .first() ,x )
Lemma190:
for G being _Graph
for W being Walk of G
for e, x being set st e Joins W .last() ,x,G holds
len (W .addEdge e) = (len W) + 2
Lemma191:
for G being _Graph
for W being Walk of G
for e, x being set st e Joins W .last() ,x,G holds
( (W .addEdge e) . ((len W) + 1) = e & (W .addEdge e) . ((len W) + 2) = x & ( for n being Element of NAT st n in dom W holds
(W .addEdge e) . n = W . n ) )
Lemma192:
for G being _Graph
for W being Walk of G
for e, x, y, z being set st W is_Walk_from x,y & e Joins y,z,G holds
W .addEdge e is_Walk_from x,z
Lemma193:
for G being _Graph
for W being Walk of G
for n being even Element of NAT st 1 <= n & n <= len W holds
( n div 2 in dom (W .edgeSeq() ) & W . n = (W .edgeSeq() ) . (n div 2) )
Lemma194:
for G being _Graph
for W being Walk of G
for n being Element of NAT holds
( n in dom (W .edgeSeq() ) iff 2 * n in dom W )
Lemma195:
for G being _Graph
for W being Walk of G ex lenWaa1 being even Element of NAT st
( lenWaa1 = (len W) - 1 & len (W .edgeSeq() ) = lenWaa1 div 2 )
Lemma196:
for G being _Graph
for W being Walk of G
for n being Element of NAT holds (W .cut 1,n) .edgeSeq() c= W .edgeSeq()
Lemma197:
for G being _Graph
for W being Walk of G
for e, x being set st e Joins W .last() ,x,G holds
(W .addEdge e) .edgeSeq() = (W .edgeSeq() ) ^ <*e*>
Lemma198:
for G being _Graph
for W being Walk of G
for x being set holds
( x in W .vertices() iff ex n being odd Element of NAT st
( n <= len W & W . n = x ) )
Lemma199:
for G being _Graph
for W being Walk of G
for e being set holds
( e in W .edges() iff ex n being even Element of NAT st
( 1 <= n & n <= len W & W . n = e ) )
Lemma201:
for G being _Graph
for W being Walk of G
for e being set st e in W .edges() holds
ex v1, v2 being Vertex of G ex n being odd Element of NAT st
( n + 2 <= len W & v1 = W . n & e = W . (n + 1) & v2 = W . (n + 2) & e Joins v1,v2,G )
Lemma202:
for G being _Graph
for W being Walk of G
for e, x, y being set st e in W .edges() & e Joins x,y,G holds
( x in W .vertices() & y in W .vertices() )
Lemma203:
for G being _Graph
for W being Walk of G holds len W = (2 * (W .length() )) + 1
by ;
Lemma204:
for G being _Graph
for W being Walk of G
for n being odd Element of NAT st n <= len W holds
W .find n <= n
Lemma205:
for G being _Graph
for W being Walk of G
for n being odd Element of NAT st n <= len W holds
W .rfind n >= n
Lemma206:
for G being _Graph
for W being Walk of G holds
( W is directed iff for n being odd Element of NAT st n < len W holds
W . (n + 1) DJoins W . n,W . (n + 2),G )
Lemma207:
for G being _Graph
for W being Walk of G
for x, e, y, z being set st W is directed & W is_Walk_from x,y & e DJoins y,z,G holds
( W .addEdge e is directed & W .addEdge e is_Walk_from x,z )
Lemma208:
for G being _Graph
for W being Walk of G
for m, n being Element of NAT st W is directed holds
W .cut m,n is directed
Lemma209:
for G being _Graph
for W being Walk of G holds
( not W is trivial iff 3 <= len W )
Lemma210:
for G being _Graph
for W being Walk of G holds
( not W is trivial iff len W <> 1 )
Lemma211:
for G being _Graph
for W being Walk of G holds
( W is trivial iff ex v being Vertex of G st W = G .walkOf v )
Lemma212:
for G being _Graph
for W being Walk of G holds
( W is Trail-like iff for m, n being even Element of NAT st 1 <= m & m < n & n <= len W holds
W . m <> W . n )
Lemma215:
for G being _Graph
for W being Walk of G holds
( W is Trail-like iff W .reverse() is Trail-like )
Lemma217:
for G being _Graph
for W being Walk of G
for m, n being Element of NAT st W is Trail-like holds
W .cut m,n is Trail-like
Lemma219:
for G being _Graph
for W being Walk of G
for e being set st W is Trail-like & e in (W .last() ) .edgesInOut() & not e in W .edges() holds
W .addEdge e is Trail-like
Lemma220:
for G being _Graph
for W being Walk of G st len W <= 3 holds
W is Trail-like
Lemma221:
for G being _Graph
for x, e, y being set st e Joins x,y,G holds
G .walkOf x,e,y is Path-like
Lemma222:
for G being _Graph
for W being Walk of G holds
( W is Path-like iff W .reverse() is Path-like )
Lemma223:
for G being _Graph
for W being Walk of G
for m, n being Element of NAT st W is Path-like holds
W .cut m,n is Path-like
Lemma224:
for G being _Graph
for W being Walk of G
for e, v being set st W is Path-like & e Joins W .last() ,v,G & not e in W .edges() & ( W is trivial or not W is closed ) & ( for n being odd Element of NAT st 1 < n & n <= len W holds
W . n <> v ) holds
W .addEdge e is Path-like
Lemma225:
for G being _Graph
for W being Walk of G st ( for m, n being odd Element of NAT st m <= len W & n <= len W & W . m = W . n holds
m = n ) holds
W is Path-like
Lemma227:
for G being _Graph
for W being Walk of G st ( for n being odd Element of NAT st n <= len W holds
W .rfind n = n ) holds
W is Path-like
Lemma228:
for G being _Graph
for W being Walk of G
for e, v being set st e Joins W .last() ,v,G & W is Path-like & not v in W .vertices() & ( W is trivial or not W is closed ) holds
W .addEdge e is Path-like
Lemma229:
for G being _Graph
for W being Walk of G st len W <= 3 holds
W is Path-like
:: deftheorem Def32 defines Subwalk GLIB_001:def 32 :
Lemma239:
for G being _Graph
for W being Walk of G holds W is Subwalk of W
Lemma240:
for G being _Graph
for W1 being Walk of G
for W2 being Subwalk of W1
for W3 being Subwalk of W2 holds W3 is Subwalk of W1
Lemma245:
for G being _Graph
for W1, W2 being Walk of G st W1 is Subwalk of W2 holds
len W1 <= len W2
:: deftheorem Def33 defines .allWalks() GLIB_001:def 33 :
:: deftheorem Def34 defines .allTrails() GLIB_001:def 34 :
:: deftheorem Def35 defines .allPaths() GLIB_001:def 35 :
:: deftheorem Def36 defines .allDWalks() GLIB_001:def 36 :
:: deftheorem Def37 defines .allDTrails() GLIB_001:def 37 :
:: deftheorem Def38 defines .allDPaths() GLIB_001:def 38 :
theorem Th7: :: GLIB_001:7
canceled;
theorem Th8: :: GLIB_001:8
theorem Th9: :: GLIB_001:9
theorem Th10: :: GLIB_001:10
theorem Th11: :: GLIB_001:11
theorem Th12: :: GLIB_001:12
theorem Th13: :: GLIB_001:13
theorem Th14: :: GLIB_001:14
theorem Th15: :: GLIB_001:15
theorem Th16: :: GLIB_001:16
for
G being
_Graph for
e,
x,
y being
set st
e Joins x,
y,
G holds
(
(G .walkOf x,e,y) .first() = x &
(G .walkOf x,e,y) .last() = y &
G .walkOf x,
e,
y is_Walk_from x,
y )
theorem Th17: :: GLIB_001:17
theorem Th18: :: GLIB_001:18
theorem Th19: :: GLIB_001:19
theorem Th20: :: GLIB_001:20
theorem Th21: :: GLIB_001:21
theorem Th22: :: GLIB_001:22
theorem Th23: :: GLIB_001:23
theorem Th24: :: GLIB_001:24
theorem Th25: :: GLIB_001:25
theorem Th26: :: GLIB_001:26
theorem Th27: :: GLIB_001:27
theorem Th28: :: GLIB_001:28
theorem Th29: :: GLIB_001:29
theorem Th30: :: GLIB_001:30
theorem Th31: :: GLIB_001:31
theorem Th32: :: GLIB_001:32
theorem Th33: :: GLIB_001:33
theorem Th34: :: GLIB_001:34
theorem Th35: :: GLIB_001:35
theorem Th36: :: GLIB_001:36
theorem Th37: :: GLIB_001:37
theorem Th38: :: GLIB_001:38
theorem Th39: :: GLIB_001:39
theorem Th40: :: GLIB_001:40
theorem Th41: :: GLIB_001:41
theorem Th42: :: GLIB_001:42
theorem Th43: :: GLIB_001:43
theorem Th44: :: GLIB_001:44
theorem Th45: :: GLIB_001:45
theorem Th46: :: GLIB_001:46
theorem Th47: :: GLIB_001:47
theorem Th48: :: GLIB_001:48
theorem Th49: :: GLIB_001:49
theorem Th50: :: GLIB_001:50
theorem Th51: :: GLIB_001:51
theorem Th52: :: GLIB_001:52
theorem Th53: :: GLIB_001:53
theorem Th54: :: GLIB_001:54
theorem Th55: :: GLIB_001:55
theorem Th56: :: GLIB_001:56
theorem Th57: :: GLIB_001:57
theorem Th58: :: GLIB_001:58
theorem Th59: :: GLIB_001:59
theorem Th60: :: GLIB_001:60
theorem Th61: :: GLIB_001:61
theorem Th62: :: GLIB_001:62
theorem Th63: :: GLIB_001:63
theorem Th64: :: GLIB_001:64
theorem Th65: :: GLIB_001:65
theorem Th66: :: GLIB_001:66
theorem Th67: :: GLIB_001:67
theorem Th68: :: GLIB_001:68
theorem Th69: :: GLIB_001:69
theorem Th70: :: GLIB_001:70
theorem Th71: :: GLIB_001:71
theorem Th72: :: GLIB_001:72
theorem Th73: :: GLIB_001:73
theorem Th74: :: GLIB_001:74
theorem Th75: :: GLIB_001:75
theorem Th76: :: GLIB_001:76
theorem Th77: :: GLIB_001:77
theorem Th78: :: GLIB_001:78
theorem Th79: :: GLIB_001:79
theorem Th80: :: GLIB_001:80
theorem Th81: :: GLIB_001:81
theorem Th82: :: GLIB_001:82
theorem Th83: :: GLIB_001:83
theorem Th84: :: GLIB_001:84
theorem Th85: :: GLIB_001:85
theorem Th86: :: GLIB_001:86
theorem Th87: :: GLIB_001:87
theorem Th88: :: GLIB_001:88
theorem Th89: :: GLIB_001:89
theorem Th90: :: GLIB_001:90
theorem Th91: :: GLIB_001:91
theorem Th92: :: GLIB_001:92
theorem Th93: :: GLIB_001:93
theorem Th94: :: GLIB_001:94
theorem Th95: :: GLIB_001:95
theorem Th96: :: GLIB_001:96
theorem Th97: :: GLIB_001:97
theorem Th98: :: GLIB_001:98
theorem Th99: :: GLIB_001:99
theorem Th100: :: GLIB_001:100
theorem Th101: :: GLIB_001:101
theorem Th102: :: GLIB_001:102
theorem Th103: :: GLIB_001:103
theorem Th104: :: GLIB_001:104
theorem Th105: :: GLIB_001:105
theorem Th106: :: GLIB_001:106
theorem Th107: :: GLIB_001:107
theorem Th108: :: GLIB_001:108
theorem Th109: :: GLIB_001:109
theorem Th110: :: GLIB_001:110
theorem Th111: :: GLIB_001:111
theorem Th112: :: GLIB_001:112
theorem Th113: :: GLIB_001:113
theorem Th114: :: GLIB_001:114
theorem Th115: :: GLIB_001:115
theorem Th116: :: GLIB_001:116
theorem Th117: :: GLIB_001:117
theorem Th118: :: GLIB_001:118
theorem Th119: :: GLIB_001:119
theorem Th120: :: GLIB_001:120
theorem Th121: :: GLIB_001:121
theorem Th122: :: GLIB_001:122
theorem Th123: :: GLIB_001:123
theorem Th124: :: GLIB_001:124
theorem Th125: :: GLIB_001:125
theorem Th126: :: GLIB_001:126
theorem Th127: :: GLIB_001:127
theorem Th128: :: GLIB_001:128
theorem Th129: :: GLIB_001:129
theorem Th130: :: GLIB_001:130
theorem Th131: :: GLIB_001:131
theorem Th132: :: GLIB_001:132
theorem Th133: :: GLIB_001:133
theorem Th134: :: GLIB_001:134
theorem Th135: :: GLIB_001:135
theorem Th136: :: GLIB_001:136
theorem Th137: :: GLIB_001:137
theorem Th138: :: GLIB_001:138
theorem Th139: :: GLIB_001:139
theorem Th140: :: GLIB_001:140
theorem Th141: :: GLIB_001:141
theorem Th142: :: GLIB_001:142
theorem Th143: :: GLIB_001:143
theorem Th144: :: GLIB_001:144
theorem Th145: :: GLIB_001:145
theorem Th146: :: GLIB_001:146
theorem Th147: :: GLIB_001:147
theorem Th148: :: GLIB_001:148
theorem Th149: :: GLIB_001:149
theorem Th150: :: GLIB_001:150
theorem Th151: :: GLIB_001:151
theorem Th152: :: GLIB_001:152
theorem Th153: :: GLIB_001:153
theorem Th154: :: GLIB_001:154
theorem Th155: :: GLIB_001:155
theorem Th156: :: GLIB_001:156
theorem Th157: :: GLIB_001:157
theorem Th158: :: GLIB_001:158
theorem Th159: :: GLIB_001:159
theorem Th160: :: GLIB_001:160
theorem Th161: :: GLIB_001:161
theorem Th162: :: GLIB_001:162
theorem Th163: :: GLIB_001:163
theorem Th164: :: GLIB_001:164
theorem Th165: :: GLIB_001:165
theorem Th166: :: GLIB_001:166
theorem Th167: :: GLIB_001:167
theorem Th168: :: GLIB_001:168
theorem Th169: :: GLIB_001:169
theorem Th170: :: GLIB_001:170
theorem Th171: :: GLIB_001:171
theorem Th172: :: GLIB_001:172
theorem Th173: :: GLIB_001:173
theorem Th174: :: GLIB_001:174
theorem Th175: :: GLIB_001:175
theorem Th176: :: GLIB_001:176
theorem Th177: :: GLIB_001:177
theorem Th178: :: GLIB_001:178
theorem Th179: :: GLIB_001:179
theorem Th180: :: GLIB_001:180
theorem Th181: :: GLIB_001:181
theorem Th182: :: GLIB_001:182