:: PASCH semantic presentation
definition
let c
1 be
OAffinSpace;
attr a
1 is
satisfying_Int_Par_Pasch means :: PASCH:def 1
for b
1, b
2, b
3, b
4, b
5 being
Element of a
1 holds
( not
LIN b
5,b
2,b
3 &
Mid b
2,b
5,b
1 &
LIN b
5,b
3,b
4 & b
2,b
3 '||' b
4,b
1 implies
Mid b
3,b
5,b
4 );
end;
:: deftheorem Def1 defines satisfying_Int_Par_Pasch PASCH:def 1 :
for b
1 being
OAffinSpace holds
( b
1 is
satisfying_Int_Par_Pasch iff for b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
( not
LIN b
6,b
3,b
4 &
Mid b
3,b
6,b
2 &
LIN b
6,b
4,b
5 & b
3,b
4 '||' b
5,b
2 implies
Mid b
4,b
6,b
5 ) );
definition
let c
1 be
OAffinSpace;
attr a
1 is
satisfying_Ext_Par_Pasch means :: PASCH:def 2
for b
1, b
2, b
3, b
4, b
5 being
Element of a
1 holds
(
Mid b
5,b
2,b
3 &
LIN b
5,b
1,b
4 & b
1,b
2 '||' b
3,b
4 & not
LIN b
5,b
1,b
2 implies
Mid b
5,b
1,b
4 );
end;
:: deftheorem Def2 defines satisfying_Ext_Par_Pasch PASCH:def 2 :
for b
1 being
OAffinSpace holds
( b
1 is
satisfying_Ext_Par_Pasch iff for b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
(
Mid b
6,b
3,b
4 &
LIN b
6,b
2,b
5 & b
2,b
3 '||' b
4,b
5 & not
LIN b
6,b
2,b
3 implies
Mid b
6,b
2,b
5 ) );
definition
let c
1 be
OAffinSpace;
attr a
1 is
satisfying_Gen_Par_Pasch means :: PASCH:def 3
for b
1, b
2, b
3, b
4, b
5, b
6 being
Element of a
1 holds
( not
LIN b
1,b
2,b
4 & b
1,b
4 '||' b
2,b
5 & b
1,b
4 '||' b
3,b
6 &
Mid b
1,b
2,b
3 &
LIN b
4,b
5,b
6 implies
Mid b
4,b
5,b
6 );
end;
:: deftheorem Def3 defines satisfying_Gen_Par_Pasch PASCH:def 3 :
for b
1 being
OAffinSpace holds
( b
1 is
satisfying_Gen_Par_Pasch 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
5 & b
2,b
5 '||' b
3,b
6 & b
2,b
5 '||' b
4,b
7 &
Mid b
2,b
3,b
4 &
LIN b
5,b
6,b
7 implies
Mid b
5,b
6,b
7 ) );
definition
let c
1 be
OAffinSpace;
attr a
1 is
satisfying_Ext_Bet_Pasch means :: PASCH:def 4
for b
1, b
2, b
3, b
4, b
5, b
6 being
Element of a
1 holds
not (
Mid b
1,b
2,b
4 &
Mid b
2,b
5,b
3 & not
LIN b
1,b
2,b
3 & ( for b
7 being
Element of a
1 holds
not (
Mid b
1,b
7,b
3 &
Mid b
7,b
5,b
4 ) ) );
end;
:: deftheorem Def4 defines satisfying_Ext_Bet_Pasch PASCH:def 4 :
for b
1 being
OAffinSpace holds
( b
1 is
satisfying_Ext_Bet_Pasch iff for b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
not (
Mid b
2,b
3,b
5 &
Mid b
3,b
6,b
4 & not
LIN b
2,b
3,b
4 & ( for b
8 being
Element of b
1 holds
not (
Mid b
2,b
8,b
4 &
Mid b
8,b
6,b
5 ) ) ) );
definition
let c
1 be
OAffinSpace;
attr a
1 is
satisfying_Int_Bet_Pasch means :: PASCH:def 5
for b
1, b
2, b
3, b
4, b
5, b
6 being
Element of a
1 holds
not (
Mid b
1,b
2,b
4 &
Mid b
1,b
5,b
3 & not
LIN b
1,b
2,b
3 & ( for b
7 being
Element of a
1 holds
not (
Mid b
2,b
7,b
3 &
Mid b
5,b
7,b
4 ) ) );
end;
:: deftheorem Def5 defines satisfying_Int_Bet_Pasch PASCH:def 5 :
for b
1 being
OAffinSpace holds
( b
1 is
satisfying_Int_Bet_Pasch iff for b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
not (
Mid b
2,b
3,b
5 &
Mid b
2,b
6,b
4 & not
LIN b
2,b
3,b
4 & ( for b
8 being
Element of b
1 holds
not (
Mid b
3,b
8,b
4 &
Mid b
6,b
8,b
5 ) ) ) );
definition
let c
1 be
OAffinSpace;
attr a
1 is
Fanoian means :: PASCH:def 6
for b
1, b
2, b
3, b
4 being
Element of a
1 holds
not ( b
1,b
2 // b
3,b
4 & b
1,b
3 // b
2,b
4 & not
LIN b
1,b
2,b
3 & ( for b
5 being
Element of a
1 holds
not (
Mid b
1,b
5,b
4 &
Mid b
2,b
5,b
3 ) ) );
end;
:: deftheorem Def6 defines Fanoian PASCH:def 6 :
for b
1 being
OAffinSpace holds
( b
1 is
Fanoian iff for b
2, b
3, b
4, b
5 being
Element of b
1 holds
not ( 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 (
Mid b
2,b
6,b
5 &
Mid b
3,b
6,b
4 ) ) ) );
theorem Th1: :: PASCH:1
canceled;
theorem Th2: :: PASCH:2
canceled;
theorem Th3: :: PASCH:3
canceled;
theorem Th4: :: PASCH:4
canceled;
theorem Th5: :: PASCH:5
canceled;
theorem Th6: :: PASCH:6
canceled;
theorem Th7: :: PASCH:7
for b
1 being
OAffinSpacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
not ( b
2,b
3 // b
3,b
4 & b
3 <> b
4 & b
2 <> b
3 & ( for b
6 being
Element of b
1 holds
not ( b
5,b
3 // b
3,b
6 & b
5,b
2 '||' b
4,b
6 & b
4 <> b
6 & b
3 <> b
6 ) ) )
theorem Th8: :: PASCH:8
for b
1 being
OAffinSpacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
not ( b
2,b
3 // b
2,b
4 & b
2 <> b
4 & b
3 <> b
2 & ( for b
6 being
Element of b
1 holds
not ( b
2,b
5 // b
2,b
6 & b
5,b
3 '||' b
4,b
6 & b
4 <> b
6 ) ) )
theorem Th9: :: PASCH:9
theorem Th10: :: PASCH:10
canceled;
theorem Th11: :: PASCH:11
for b
1 being
OAffinSpacefor 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 &
LIN b
2,b
4,b
5 &
LIN b
2,b
3,b
6 &
LIN b
2,b
3,b
7 & b
3,b
4 '||' b
5,b
6 & b
3,b
4 '||' b
5,b
7 implies b
6 = b
7 )
theorem Th12: :: PASCH:12
for b
1 being
OAffinSpacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
( not
LIN b
2,b
3,b
4 & b
2,b
3 '||' b
4,b
5 & b
2,b
3 '||' b
4,b
6 & b
2,b
4 '||' b
3,b
5 & b
2,b
4 '||' b
3,b
6 implies b
5 = b
6 )
theorem Th13: :: PASCH:13
for b
1 being
OAffinSpacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
( not
LIN b
2,b
3,b
4 &
Mid b
3,b
2,b
5 &
LIN b
2,b
4,b
6 & b
3,b
4 '||' b
6,b
5 implies
Mid b
4,b
2,b
6 )
theorem Th14: :: PASCH:14
theorem Th15: :: PASCH:15
for b
1 being
OAffinSpacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
(
Mid b
2,b
3,b
4 &
LIN b
2,b
5,b
6 & b
5,b
3 '||' b
4,b
6 & not
LIN b
2,b
5,b
3 implies
Mid b
2,b
5,b
6 )
theorem Th16: :: PASCH:16
theorem Th17: :: PASCH:17
for b
1 being
OAffinSpacefor 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 & b
2,b
4 '||' b
3,b
5 & b
2,b
4 '||' b
6,b
7 &
Mid b
2,b
3,b
6 &
LIN b
4,b
5,b
7 implies
Mid b
4,b
5,b
7 )
theorem Th18: :: PASCH:18
theorem Th19: :: PASCH:19
for b
1 being
OAffinSpacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
( not
LIN b
2,b
3,b
4 & b
3,b
2 // b
2,b
5 & b
4,b
2 // b
2,b
6 & b
3,b
4 '||' b
5,b
6 implies b
3,b
4 // b
6,b
5 )
theorem Th20: :: PASCH:20
for b
1 being
OAffinSpacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
( not
LIN b
2,b
3,b
4 & b
2,b
3 // b
2,b
5 & b
2,b
4 // b
2,b
6 & b
3,b
4 '||' b
5,b
6 implies b
3,b
4 // b
5,b
6 )
theorem Th21: :: PASCH:21
for b
1 being
OAffinSpacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
( not
LIN b
2,b
3,b
4 & b
2,b
3 '||' b
4,b
5 & b
2,b
4 '||' b
3,b
5 implies ( b
2,b
3 // b
4,b
5 & b
2,b
4 // b
3,b
5 ) )
theorem Th22: :: PASCH:22
for b
1 being
OAffinSpacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
(
Mid b
2,b
3,b
4 & b
3,b
5 // b
4,b
6 & b
2,b
5 // b
2,b
6 & not
LIN b
2,b
6,b
4 & b
2 <> b
3 implies
Mid b
2,b
5,b
6 )
theorem Th23: :: PASCH:23
for b
1 being
OAffinSpacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
(
Mid b
2,b
3,b
4 & b
5,b
3 // b
6,b
4 & b
2,b
5 // b
2,b
6 & not
LIN b
2,b
4,b
6 & b
2 <> b
5 implies
Mid b
2,b
5,b
6 )
theorem Th24: :: PASCH:24
for b
1 being
OAffinSpacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
not ( not
LIN b
2,b
3,b
4 & b
2,b
4 // b
2,b
5 & b
4,b
3 // b
5,b
6 &
LIN b
3,b
2,b
6 & b
2 <> b
6 &
Mid b
3,b
2,b
6 )
theorem Th25: :: PASCH:25
for b
1 being
OAffinSpacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
not ( b
2,b
3 // b
2,b
4 & b
3 <> b
2 & ( for b
6 being
Element of b
1 holds
not ( b
2,b
5 // b
2,b
6 & b
3,b
5 // b
4,b
6 ) ) )
theorem Th26: :: PASCH:26
for b
1 being
OAffinSpacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
not (
Mid b
2,b
3,b
4 & ( for b
6 being
Element of b
1 holds
not (
Mid b
2,b
6,b
5 & b
4,b
5 // b
3,b
6 ) ) )
theorem Th27: :: PASCH:27
for b
1 being
OAffinSpacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
not ( b
2 <> b
3 &
Mid b
2,b
3,b
4 & ( for b
6 being
Element of b
1 holds
not (
Mid b
2,b
5,b
6 & b
3,b
5 // b
4,b
6 ) ) )
theorem Th28: :: PASCH:28
for b
1 being
OAffinSpacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
not ( not
LIN b
2,b
3,b
4 &
Mid b
2,b
5,b
4 & ( for b
6 being
Element of b
1 holds
not (
Mid b
2,b
6,b
3 & b
3,b
4 // b
6,b
5 ) ) )
theorem Th29: :: PASCH:29
theorem Th30: :: PASCH:30
for b
1 being
OAffinSpacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
not ( b
2,b
3 // b
4,b
5 & not
LIN b
2,b
3,b
4 & ( for b
6 being
Element of b
1 holds
not (
Mid b
2,b
6,b
5 &
Mid b
3,b
6,b
4 ) ) )
theorem Th31: :: PASCH:31
canceled;
theorem Th32: :: PASCH:32
theorem Th33: :: PASCH:33
for b
1 being
OAffinSpacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
not ( 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
6,b
2,b
5 &
LIN b
6,b
3,b
4 ) ) )
theorem Th34: :: PASCH:34
for b
1 being
OAffinSpacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
not ( b
2,b
3 '||' b
4,b
5 & b
2,b
4 '||' b
3,b
5 & not
LIN b
2,b
3,b
4 &
LIN b
6,b
2,b
5 &
LIN b
6,b
3,b
4 &
LIN b
6,b
2,b
3 )
theorem Th35: :: PASCH:35
for b
1 being
OAffinSpacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
not (
Mid b
2,b
3,b
4 &
Mid b
3,b
5,b
6 & not
LIN b
2,b
3,b
6 & ( for b
7 being
Element of b
1 holds
not (
Mid b
2,b
7,b
6 &
Mid b
7,b
5,b
4 ) ) )
theorem Th36: :: PASCH:36
theorem Th37: :: PASCH:37
for b
1 being
OAffinSpacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
not (
Mid b
2,b
3,b
4 &
Mid b
2,b
5,b
6 & not
LIN b
2,b
3,b
6 & ( for b
7 being
Element of b
1 holds
not (
Mid b
3,b
7,b
6 &
Mid b
5,b
7,b
4 ) ) )
theorem Th38: :: PASCH:38
theorem Th39: :: PASCH:39
for b
1 being
OAffinSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
(
Mid b
2,b
3,b
4 & b
2,b
3 // b
5,b
6 & not
LIN b
2,b
3,b
5 &
LIN b
5,b
6,b
7 & b
2,b
5 // b
3,b
6 & b
2,b
5 // b
4,b
7 implies
Mid b
5,b
6,b
7 )