:: TDGROUP semantic presentation
theorem Th1: :: TDGROUP:1
canceled;
theorem Th2: :: TDGROUP:2
theorem Th3: :: TDGROUP:3
:: deftheorem Def1 defines Two_Divisible TDGROUP:def 1 :
Lemma4:
G_Real is Fanoian
theorem Th4: :: TDGROUP:4
canceled;
theorem Th5: :: TDGROUP:5
canceled;
theorem Th6: :: TDGROUP:6
canceled;
theorem Th7: :: TDGROUP:7
definition
let c
1 be non
empty LoopStr ;
canceled;canceled;func CONGRD c
1 -> Relation of
[:the carrier of a1,the carrier of a1:] means :
Def4:
:: TDGROUP:def 4
for b
1, b
2, b
3, b
4 being
Element of a
1 holds
(
[[b1,b2],[b3,b4]] in a
2 iff b
1 # b
4 = b
2 # b
3 );
existence
ex b1 being Relation of [:the carrier of c1,the carrier of c1:] st
for b2, b3, b4, b5 being Element of c1 holds
( [[b2,b3],[b4,b5]] in b1 iff b2 # b5 = b3 # b4 )
uniqueness
for b1, b2 being Relation of [:the carrier of c1,the carrier of c1:] holds
( ( for b3, b4, b5, b6 being Element of c1 holds
( [[b3,b4],[b5,b6]] in b1 iff b3 # b6 = b4 # b5 ) ) & ( for b3, b4, b5, b6 being Element of c1 holds
( [[b3,b4],[b5,b6]] in b2 iff b3 # b6 = b4 # b5 ) ) implies b1 = b2 )
end;
:: deftheorem Def2 TDGROUP:def 2 :
canceled;
:: deftheorem Def3 TDGROUP:def 3 :
canceled;
:: deftheorem Def4 defines CONGRD TDGROUP:def 4 :
:: deftheorem Def5 defines AV TDGROUP:def 5 :
theorem Th8: :: TDGROUP:8
canceled;
theorem Th9: :: TDGROUP:9
:: deftheorem Def6 defines ==> TDGROUP:def 6 :
theorem Th10: :: TDGROUP:10
theorem Th11: :: TDGROUP:11
theorem Th12: :: TDGROUP:12
theorem Th13: :: TDGROUP:13
theorem Th14: :: TDGROUP:14
for b
1 being
Uniquely_Two_Divisible_Groupfor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2,b
3 ==> b
4,b
5 & b
6,b
7 ==> b
4,b
5 implies b
2,b
3 ==> b
6,b
7 )
theorem Th15: :: TDGROUP:15
theorem Th16: :: TDGROUP:16
for b
1 being
Uniquely_Two_Divisible_Groupfor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2,b
3 ==> b
4,b
5 & b
2,b
6 ==> b
4,b
7 implies b
3,b
6 ==> b
5,b
7 )
theorem Th17: :: TDGROUP:17
theorem Th18: :: TDGROUP:18
theorem Th19: :: TDGROUP:19
theorem Th20: :: TDGROUP:20
for b
1 being
Uniquely_Two_Divisible_Group holds
( not for b
2, b
3 being
Element of b
1 holds not b
2 <> b
3 implies ( not for b
2, b
3 being
Element of
(AV b1) holds not b
2 <> b
3 & ( for b
2, b
3, b
4 being
Element of
(AV b1) holds
( b
2,b
3 // b
4,b
4 implies b
2 = b
3 ) ) & ( for b
2, b
3, b
4, b
5, b
6, b
7 being
Element of
(AV b1) holds
( b
2,b
3 // b
6,b
7 & b
4,b
5 // b
6,b
7 implies b
2,b
3 // b
4,b
5 ) ) & ( for b
2, b
3, b
4 being
Element of
(AV b1) holds
ex b
5 being
Element of
(AV b1) st b
2,b
3 // b
4,b
5 ) & ( for b
2, b
3, b
4, b
5, b
6, b
7 being
Element of
(AV b1) holds
( b
2,b
3 // b
5,b
6 & b
2,b
4 // b
5,b
7 implies b
3,b
4 // b
6,b
7 ) ) & ( for b
2, b
3 being
Element of
(AV b1) holds
ex b
4 being
Element of
(AV b1) st b
2,b
4 // b
4,b
3 ) & ( for b
2, b
3, b
4, b
5 being
Element of
(AV b1) holds
( b
2,b
3 // b
3,b
4 & b
2,b
5 // b
5,b
4 implies b
3 = b
5 ) ) & ( for b
2, b
3, b
4, b
5 being
Element of
(AV b1) holds
( b
2,b
3 // b
4,b
5 implies b
2,b
4 // b
3,b
5 ) ) ) )
definition
let c
1 be non
empty AffinStruct ;
canceled;attr a
1 is
AffVect-like means :
Def8:
:: TDGROUP:def 8
( ( for b
1, b
2, b
3 being
Element of a
1 holds
( b
1,b
2 // b
3,b
3 implies b
1 = b
2 ) ) & ( for b
1, b
2, b
3, b
4, b
5, b
6 being
Element of a
1 holds
( b
1,b
2 // b
5,b
6 & b
3,b
4 // b
5,b
6 implies b
1,b
2 // b
3,b
4 ) ) & ( for b
1, b
2, b
3 being
Element of a
1 holds
ex b
4 being
Element of a
1 st b
1,b
2 // b
3,b
4 ) & ( for b
1, b
2, b
3, b
4, b
5, b
6 being
Element of a
1 holds
( b
1,b
2 // b
4,b
5 & b
1,b
3 // b
4,b
6 implies b
2,b
3 // b
5,b
6 ) ) & ( for b
1, b
2 being
Element of a
1 holds
ex b
3 being
Element of a
1 st b
1,b
3 // b
3,b
2 ) & ( for b
1, b
2, b
3, b
4 being
Element of a
1 holds
( b
1,b
2 // b
2,b
3 & b
1,b
4 // b
4,b
3 implies b
2 = b
4 ) ) & ( for b
1, b
2, b
3, b
4 being
Element of a
1 holds
( b
1,b
2 // b
3,b
4 implies b
1,b
3 // b
2,b
4 ) ) );
end;
:: deftheorem Def7 TDGROUP:def 7 :
canceled;
:: deftheorem Def8 defines AffVect-like TDGROUP:def 8 :
for b
1 being non
empty AffinStruct holds
( b
1 is
AffVect-like iff ( ( for b
2, b
3, b
4 being
Element of b
1 holds
( b
2,b
3 // b
4,b
4 implies b
2 = b
3 ) ) & ( for b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2,b
3 // b
6,b
7 & b
4,b
5 // b
6,b
7 implies b
2,b
3 // b
4,b
5 ) ) & ( for b
2, b
3, b
4 being
Element of b
1 holds
ex b
5 being
Element of b
1 st b
2,b
3 // b
4,b
5 ) & ( for b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2,b
3 // b
5,b
6 & b
2,b
4 // b
5,b
7 implies b
3,b
4 // b
6,b
7 ) ) & ( for b
2, b
3 being
Element of b
1 holds
ex b
4 being
Element of b
1 st b
2,b
4 // b
4,b
3 ) & ( for b
2, b
3, b
4, b
5 being
Element of b
1 holds
( b
2,b
3 // b
3,b
4 & b
2,b
5 // b
5,b
4 implies b
3 = b
5 ) ) & ( for b
2, b
3, b
4, b
5 being
Element of b
1 holds
( b
2,b
3 // b
4,b
5 implies b
2,b
4 // b
3,b
5 ) ) ) );
theorem Th21: :: TDGROUP:21
for b
1 being non
empty AffinStruct holds
( ( not for b
2, b
3 being
Element of b
1 holds not b
2 <> b
3 & ( for b
2, b
3, b
4 being
Element of b
1 holds
( b
2,b
3 // b
4,b
4 implies b
2 = b
3 ) ) & ( for b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2,b
3 // b
6,b
7 & b
4,b
5 // b
6,b
7 implies b
2,b
3 // b
4,b
5 ) ) & ( for b
2, b
3, b
4 being
Element of b
1 holds
ex b
5 being
Element of b
1 st b
2,b
3 // b
4,b
5 ) & ( for b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2,b
3 // b
5,b
6 & b
2,b
4 // b
5,b
7 implies b
3,b
4 // b
6,b
7 ) ) & ( for b
2, b
3 being
Element of b
1 holds
ex b
4 being
Element of b
1 st b
2,b
4 // b
4,b
3 ) & ( for b
2, b
3, b
4, b
5 being
Element of b
1 holds
( b
2,b
3 // b
3,b
4 & b
2,b
5 // b
5,b
4 implies b
3 = b
5 ) ) & ( for b
2, b
3, b
4, b
5 being
Element of b
1 holds
( b
2,b
3 // b
4,b
5 implies b
2,b
4 // b
3,b
5 ) ) ) iff b
1 is
AffVect )
by Def8, REALSET2:def 7;
theorem Th22: :: TDGROUP:22