:: PENCIL_3 semantic presentation
theorem Th1: :: PENCIL_3:1
theorem Th2: :: PENCIL_3:2
theorem Th3: :: PENCIL_3:3
theorem Th4: :: PENCIL_3:4
theorem Th5: :: PENCIL_3:5
theorem Th6: :: PENCIL_3:6
theorem Th7: :: PENCIL_3:7
theorem Th8: :: PENCIL_3:8
theorem Th9: :: PENCIL_3:9
theorem Th10: :: PENCIL_3:10
theorem Th11: :: PENCIL_3:11
theorem Th12: :: PENCIL_3:12
theorem Th13: :: PENCIL_3:13
theorem Th14: :: PENCIL_3:14
theorem Th15: :: PENCIL_3:15
theorem Th16: :: PENCIL_3:16
theorem Th17: :: PENCIL_3:17
theorem Th18: :: PENCIL_3:18
:: deftheorem Def1 defines diff PENCIL_3:def 1 :
theorem Th19: :: PENCIL_3:19
:: deftheorem Def2 defines '||' PENCIL_3:def 2 :
theorem Th20: :: PENCIL_3:20
theorem Th21: :: PENCIL_3:21
theorem Th22: :: PENCIL_3:22
theorem Th23: :: PENCIL_3:23
theorem Th24: :: PENCIL_3:24
theorem Th25: :: PENCIL_3:25
definition
let c
1 be non
empty finite set ;
let c
2 be
PLS-yielding ManySortedSet of c
1;
assume E26:
for b
1 being
Element of c
1 holds c
2 . b
1 is
strongly_connected
;
let c
3 be
Collineation of
(Segre_Product c2);
func permutation_of_indices c
3 -> Permutation of a
1 means :
Def3:
:: PENCIL_3:def 3
for b
1, b
2 being
Element of a
1 holds
( a
4 . b
1 = b
2 iff for b
3 being
Segre-Coset of a
2for b
4, b
5 being non
trivial-yielding Segre-like ManySortedSubset of
Carrier a
2 holds
( b
3 = product b
4 & a
3 .: b
3 = product b
5 &
indx b
4 = b
1 implies
indx b
5 = b
2 ) );
existence
ex b1 being Permutation of c1 st
for b2, b3 being Element of c1 holds
( b1 . b2 = b3 iff for b4 being Segre-Coset of c2
for b5, b6 being non trivial-yielding Segre-like ManySortedSubset of Carrier c2 holds
( b4 = product b5 & c3 .: b4 = product b6 & indx b5 = b2 implies indx b6 = b3 ) )
by E26, Th25;
uniqueness
for b1, b2 being Permutation of c1 holds
( ( for b3, b4 being Element of c1 holds
( b1 . b3 = b4 iff for b5 being Segre-Coset of c2
for b6, b7 being non trivial-yielding Segre-like ManySortedSubset of Carrier c2 holds
( b5 = product b6 & c3 .: b5 = product b7 & indx b6 = b3 implies indx b7 = b4 ) ) ) & ( for b3, b4 being Element of c1 holds
( b2 . b3 = b4 iff for b5 being Segre-Coset of c2
for b6, b7 being non trivial-yielding Segre-like ManySortedSubset of Carrier c2 holds
( b5 = product b6 & c3 .: b5 = product b7 & indx b6 = b3 implies indx b7 = b4 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def3 defines permutation_of_indices PENCIL_3:def 3 :
:: deftheorem Def4 defines canonical_embedding PENCIL_3:def 4 :
theorem Th26: :: PENCIL_3:26
theorem Th27: :: PENCIL_3:27
:: deftheorem Def5 defines canonical_embedding PENCIL_3:def 5 :
theorem Th28: :: PENCIL_3:28