:: PAPDESAF semantic presentation
theorem Th1: :: PAPDESAF:1
canceled;
theorem Th2: :: PAPDESAF:2
theorem Th3: :: PAPDESAF:3
theorem Th4: :: PAPDESAF:4
theorem Th5: :: PAPDESAF:5
for b
1 being
RealLinearSpacefor b
2 being
OAffinSpace holds
( b
2 = OASpace b
1 implies for b
3, b
4, b
5, b
6 being
Element of b
2for b
7, b
8, b
9, b
10 being
VECTOR of b
1 holds
( b
3 = b
7 & b
4 = b
8 & b
5 = b
9 & b
6 = b
10 implies ( b
3,b
4 '||' b
5,b
6 iff b
7,b
8 '||' b
9,b
10 ) ) )
theorem Th6: :: PAPDESAF:6
definition
let c
1 be
AffinSpace;
canceled;canceled;canceled;canceled;redefine attr a
1 is
Fanoian means :
Def5:
:: PAPDESAF:def 5
for b
1, b
2, b
3, b
4 being
Element of a
1 holds
( b
1,b
2 // b
3,b
4 & b
1,b
3 // b
2,b
4 & b
1,b
4 // b
2,b
3 implies b
1,b
2 // b
1,b
3 );
compatibility
( c1 satisfies_Fano iff for b1, b2, b3, b4 being Element of c1 holds
( b1,b2 // b3,b4 & b1,b3 // b2,b4 & b1,b4 // b2,b3 implies b1,b2 // b1,b3 ) )
end;
:: deftheorem Def1 PAPDESAF:def 1 :
canceled;
:: deftheorem Def2 PAPDESAF:def 2 :
canceled;
:: deftheorem Def3 PAPDESAF:def 3 :
canceled;
:: deftheorem Def4 PAPDESAF:def 4 :
canceled;
:: deftheorem Def5 defines satisfies_Fano PAPDESAF:def 5 :
for b
1 being
AffinSpace holds
( b
1 satisfies_Fano iff for b
2, b
3, b
4, b
5 being
Element of b
1 holds
( b
2,b
3 // b
4,b
5 & b
2,b
4 // b
3,b
5 & b
2,b
5 // b
3,b
4 implies b
2,b
3 // b
2,b
4 ) );
:: deftheorem Def6 PAPDESAF:def 6 :
canceled;
:: deftheorem Def7 PAPDESAF:def 7 :
canceled;
:: deftheorem Def8 PAPDESAF:def 8 :
canceled;
:: deftheorem Def9 PAPDESAF:def 9 :
canceled;
:: deftheorem Def10 PAPDESAF:def 10 :
canceled;
:: deftheorem Def11 defines Pappian PAPDESAF:def 11 :
:: deftheorem Def12 defines Desarguesian PAPDESAF:def 12 :
:: deftheorem Def13 defines Moufangian PAPDESAF:def 13 :
:: deftheorem Def14 defines translation PAPDESAF:def 14 :
definition
let c
1 be
OAffinSpace;
attr a
1 is
satisfying_DES means :
Def15:
:: PAPDESAF:def 15
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
Element of a
1 holds
( b
1,b
2 // b
1,b
5 & b
1,b
3 // b
1,b
6 & b
1,b
4 // b
1,b
7 & not
LIN b
1,b
2,b
3 & not
LIN b
1,b
2,b
4 & 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 );
end;
:: deftheorem Def15 defines satisfying_DES PAPDESAF:def 15 :
for b
1 being
OAffinSpace holds
( b
1 is
satisfying_DES iff for b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of b
1 holds
( b
2,b
3 // b
2,b
6 & b
2,b
4 // b
2,b
7 & b
2,b
5 // b
2,b
8 & not
LIN b
2,b
3,b
4 & not
LIN b
2,b
3,b
5 & b
3,b
4 // b
6,b
7 & b
3,b
5 // b
6,b
8 implies b
4,b
5 // b
7,b
8 ) );
definition
let c
1 be
OAffinSpace;
attr a
1 is
satisfying_DES_1 means :
Def16:
:: PAPDESAF:def 16
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
Element of a
1 holds
( b
2,b
1 // b
1,b
5 & b
3,b
1 // b
1,b
6 & b
4,b
1 // b
1,b
7 & not
LIN b
1,b
2,b
3 & not
LIN b
1,b
2,b
4 & b
2,b
3 // b
6,b
5 & b
2,b
4 // b
7,b
5 implies b
3,b
4 // b
7,b
6 );
end;
:: deftheorem Def16 defines satisfying_DES_1 PAPDESAF:def 16 :
for b
1 being
OAffinSpace holds
( b
1 is
satisfying_DES_1 iff for b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of b
1 holds
( b
3,b
2 // b
2,b
6 & b
4,b
2 // b
2,b
7 & b
5,b
2 // b
2,b
8 & not
LIN b
2,b
3,b
4 & not
LIN b
2,b
3,b
5 & b
3,b
4 // b
7,b
6 & b
3,b
5 // b
8,b
6 implies b
4,b
5 // b
8,b
7 ) );
theorem Th7: :: PAPDESAF:7
canceled;
theorem Th8: :: PAPDESAF:8
canceled;
theorem Th9: :: PAPDESAF:9
canceled;
theorem Th10: :: PAPDESAF:10
canceled;
theorem Th11: :: PAPDESAF:11
theorem Th12: :: PAPDESAF:12
for b
1 being
OAffinSpacefor b
2, b
3, b
4, b
5, b
6 being
Element of the
carrier of b
1 holds
( not
LIN b
2,b
3,b
4 & b
3,b
2 // b
2,b
5 &
LIN b
2,b
4,b
6 & b
3,b
4 '||' b
5,b
6 implies ( b
4,b
2 // b
2,b
6 & b
3,b
4 // b
6,b
5 ) )
theorem Th13: :: PAPDESAF:13
for b
1 being
OAffinSpacefor b
2, b
3, b
4, b
5, b
6 being
Element of the
carrier of b
1 holds
( not
LIN b
2,b
3,b
4 & b
2,b
3 // b
2,b
5 &
LIN b
2,b
4,b
6 & b
3,b
4 '||' b
5,b
6 implies ( b
2,b
4 // b
2,b
6 & b
3,b
4 // b
5,b
6 ) )
theorem Th14: :: PAPDESAF:14
theorem Th15: :: PAPDESAF:15
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6 being
VECTOR of b
1for b
7 being
Real holds
( b
2 - b
3 = b
7 * (b5 - b2) & b
7 <> 0 & b
2,b
4 '||' b
2,b
6 & not b
2,b
3 '||' b
2,b
4 & b
3,b
4 '||' b
5,b
6 implies ( b
6 = b
5 + (((- b7) " ) * (b4 - b3)) & b
6 = b
2 + (((- b7) " ) * (b4 - b2)) & b
4 - b
3 = (- b7) * (b6 - b5) ) )
Lemma17:
for b1 being RealLinearSpace
for b2, b3, b4 being VECTOR of b1 holds
not ( b2 <> b3 & b4 <> b3 & b2,b3 // b3,b4 & ( for b5 being Real holds
not ( b3 - b2 = b5 * (b4 - b3) & 0 < b5 ) ) )
theorem Th16: :: PAPDESAF:16
canceled;
theorem Th17: :: PAPDESAF:17
theorem Th18: :: PAPDESAF:18
Lemma20:
for b1 being RealLinearSpace
for b2, b3, b4 being VECTOR of b1 holds
not ( b2,b3 '||' b2,b4 & b2 <> b3 & b2 <> b4 & ( for b5 being Real holds
not ( b3 - b2 = b5 * (b4 - b2) & b5 <> 0 ) ) )
Lemma21:
for b1 being RealLinearSpace
for b2, b3, b4 being VECTOR of b1 holds (b2 - b4) - (b3 - b4) = b2 - b3
theorem Th19: :: PAPDESAF:19
theorem Th20: :: PAPDESAF:20
theorem Th21: :: PAPDESAF:21
theorem Th22: :: PAPDESAF:22
theorem Th23: :: PAPDESAF:23
theorem Th24: :: PAPDESAF:24