:: TRANSLAC semantic presentation
definition
let c
1 be
AffinSpace;
attr a
1 is
Fanoian means :
Def1:
:: TRANSLAC:def 1
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
LIN b
1,b
2,b
3 );
end;
:: deftheorem Def1 defines Fanoian TRANSLAC:def 1 :
for b
1 being
AffinSpace holds
( b
1 is
Fanoian 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
LIN b
2,b
3,b
4 ) );
Lemma2:
for b1 being AffinSpace
for b2, b3, b4, b5 being Element of b1 holds
not ( LIN b2,b3,b4 & b2 <> b3 & b2 <> b4 & b3 <> b4 & not LIN b2,b3,b5 & ( for b6 being Element of b1 holds
not ( LIN b5,b2,b6 & b5 <> b6 & b2 <> b6 ) ) )
Lemma3:
for b1 being AffinSpace
for b2, b3, b4, b5, b6 being Element of b1 holds
not ( LIN b2,b3,b4 & b2 <> b3 & b2 <> b4 & b3 <> b4 & not LIN b2,b3,b5 & LIN b2,b3,b6 & ( for b7 being Element of b1 holds
not ( LIN b5,b6,b7 & b5 <> b7 & b6 <> b7 ) ) )
Lemma4:
for b1 being AffinSpace
for b2, b3, b4, b5, b6 being Element of b1 holds
not ( LIN b2,b3,b4 & b2 <> b3 & b2 <> b4 & b3 <> b4 & not LIN b2,b3,b5 & b5 <> b6 & b5,b6 // b2,b3 & ( for b7 being Element of b1 holds
not ( LIN b5,b6,b7 & b5 <> b7 & b6 <> b7 ) ) )
theorem Th1: :: TRANSLAC:1
canceled;
theorem Th2: :: TRANSLAC:2
for b
1 being
AffinSpace holds
( ex b
2, b
3, b
4 being
Element of b
1 st
(
LIN b
2,b
3,b
4 & b
2 <> b
3 & b
2 <> b
4 & b
3 <> b
4 ) implies for b
2, b
3 being
Element of b
1 holds
not ( b
2 <> b
3 & ( for b
4 being
Element of b
1 holds
not (
LIN b
2,b
3,b
4 & b
2 <> b
4 & b
3 <> b
4 ) ) ) )
theorem Th3: :: TRANSLAC:3
canceled;
theorem Th4: :: TRANSLAC:4
for b
1 being
AffinPlanefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
not ( b
1 satisfies_Fano & b
2,b
3 // b
4,b
5 & b
2,b
4 // b
3,b
5 & not
LIN b
2,b
3,b
4 & ( for b
6 being
Element of b
1 holds
not (
LIN b
3,b
4,b
6 &
LIN b
2,b
5,b
6 ) ) )
Lemma7:
for b1 being AffinPlane
for b2, b3, b4, b5 being Element of b1 holds
( not LIN b2,b3,b4 & b2,b3 // b4,b5 & b4 <> b5 implies ( not LIN b4,b5,b2 & not LIN b3,b2,b5 & not LIN b5,b4,b3 ) )
Lemma8:
for b1 being AffinPlane
for b2, b3, b4, b5 being Element of b1 holds
( not LIN b2,b3,b4 & b2,b3 // b4,b5 & b2,b4 // b3,b5 implies ( not LIN b4,b5,b2 & not LIN b3,b2,b5 & not LIN b5,b4,b3 ) )
theorem Th5: :: TRANSLAC:5
theorem Th6: :: TRANSLAC:6
for b
1 being
AffinPlane holds
( b
1 satisfies_des iff for b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( not
LIN b
2,b
3,b
4 & not
LIN b
2,b
3,b
5 & b
2,b
3 // b
4,b
6 & b
2,b
3 // b
5,b
7 & b
2,b
4 // b
3,b
6 & b
2,b
5 // b
3,b
7 implies b
4,b
5 // b
6,b
7 ) )
theorem Th7: :: TRANSLAC:7
Lemma12:
for b1 being AffinPlane
for b2, b3, b4 being Element of b1 holds
not ( b2 <> b3 & ( for b5 being Element of b1 holds
( not ( not LIN b2,b3,b4 & b2,b3 // b4,b5 & b2,b4 // b3,b5 ) & not ( LIN b2,b3,b4 & ex b6, b7 being Element of b1 st
( not LIN b2,b3,b6 & b2,b3 // b6,b7 & b2,b6 // b3,b7 & b6,b7 // b4,b5 & b6,b4 // b7,b5 ) ) ) ) )
Lemma13:
for b1 being AffinPlane
for b2, b3, b4 being Element of b1 holds
not ( b2 <> b3 & ( for b5 being Element of b1 holds
( not ( not LIN b2,b3,b5 & b2,b3 // b5,b4 & b2,b5 // b3,b4 ) & not ( LIN b2,b3,b5 & ex b6, b7 being Element of b1 st
( not LIN b2,b3,b6 & b2,b3 // b6,b7 & b2,b6 // b3,b7 & b6,b7 // b5,b4 & b6,b5 // b7,b4 ) ) ) ) )
Lemma14:
for b1 being AffinPlane
for b2, b3, b4, b5, b6, b7, b8, b9, b10 being Element of b1 holds
( b1 satisfies_des & b2 <> b3 & LIN b2,b3,b4 & b2,b3 // b5,b6 & b2,b3 // b7,b8 & b2,b5 // b3,b6 & b2,b7 // b3,b8 & b5,b6 // b4,b9 & b7,b8 // b4,b10 & b5,b4 // b6,b9 & b7,b4 // b8,b10 & not LIN b2,b3,b5 & not LIN b2,b3,b7 & not LIN b5,b6,b7 implies b9 = b10 )
theorem Th8: :: TRANSLAC:8
for b
1 being
AffinPlanefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
( ( for b
6, b
7, b
8 being
Element of b
1 holds
not ( b
6 <> b
7 &
LIN b
6,b
7,b
8 & not b
8 = b
6 & not b
8 = b
7 ) ) & b
2,b
3 // b
4,b
5 & b
2,b
4 // b
3,b
5 & not
LIN b
2,b
3,b
4 implies b
2,b
5 // b
3,b
4 )
Lemma16:
for b1 being AffinPlane
for b2, b3, b4, b5, b6, b7, b8, b9, b10 being Element of b1 holds
( b1 satisfies_des & b2 <> b3 & LIN b2,b3,b4 & b2,b3 // b5,b6 & b2,b3 // b7,b8 & b2,b5 // b3,b6 & b2,b7 // b3,b8 & b5,b6 // b4,b9 & b7,b8 // b4,b10 & b5,b4 // b6,b9 & b7,b4 // b8,b10 & not LIN b2,b3,b5 & not LIN b2,b3,b7 implies b9 = b10 )
Lemma17:
for b1 being AffinPlane
for b2, b3, b4, b5, b6, b7 being Element of b1 holds
( b2 <> b3 & LIN b2,b3,b4 & b2,b3 // b5,b6 & b2,b5 // b3,b6 & b5,b6 // b4,b7 & not LIN b2,b3,b5 implies ( b5 <> b6 & LIN b2,b3,b7 ) )
Lemma18:
for b1 being AffinPlane
for b2, b3, b4, b5, b6, b7, b8, b9, b10 being Element of b1 holds
( b1 satisfies_des & b2 <> b3 & LIN b2,b3,b4 & b2,b3 // b5,b6 & b2,b3 // b7,b8 & b2,b5 // b3,b6 & b2,b7 // b3,b8 & b5,b6 // b4,b9 & b5,b4 // b6,b9 & b7,b8 // b10,b9 & b7,b10 // b8,b9 & not LIN b2,b3,b5 & not LIN b2,b3,b7 implies b4 = b10 )
Lemma19:
for b1 being AffinPlane
for b2, b3 being Element of b1 holds
not ( b1 satisfies_des & b2 <> b3 & ( for b4 being Permutation of the carrier of b1 holds
not ( b4 is_Tr & b4 . b2 = b3 ) ) )
theorem Th9: :: TRANSLAC:9
theorem Th10: :: TRANSLAC:10
theorem Th11: :: TRANSLAC:11
theorem Th12: :: TRANSLAC:12
theorem Th13: :: TRANSLAC:13
theorem Th14: :: TRANSLAC:14
theorem Th15: :: TRANSLAC:15
theorem Th16: :: TRANSLAC:16