:: PASCH semantic presentation

definition
let c1 be OAffinSpace;
attr a1 is satisfying_Int_Par_Pasch means :: PASCH:def 1
for b1, b2, b3, b4, b5 being Element of a1 holds
( not LIN b5,b2,b3 & Mid b2,b5,b1 & LIN b5,b3,b4 & b2,b3 '||' b4,b1 implies Mid b3,b5,b4 );
end;

:: deftheorem Def1 defines satisfying_Int_Par_Pasch PASCH:def 1 :
for b1 being OAffinSpace holds
( b1 is satisfying_Int_Par_Pasch iff for b2, b3, b4, b5, b6 being Element of b1 holds
( not LIN b6,b3,b4 & Mid b3,b6,b2 & LIN b6,b4,b5 & b3,b4 '||' b5,b2 implies Mid b4,b6,b5 ) );

notation
let c1 be OAffinSpace;
synonym c1 satisfies_Int_Par_Pasch for satisfying_Int_Par_Pasch c1c1 satisfies_Int_Par_Paschsatisfies_Int_Par_Pasch ;
end;

definition
let c1 be OAffinSpace;
attr a1 is satisfying_Ext_Par_Pasch means :: PASCH:def 2
for b1, b2, b3, b4, b5 being Element of a1 holds
( Mid b5,b2,b3 & LIN b5,b1,b4 & b1,b2 '||' b3,b4 & not LIN b5,b1,b2 implies Mid b5,b1,b4 );
end;

:: deftheorem Def2 defines satisfying_Ext_Par_Pasch PASCH:def 2 :
for b1 being OAffinSpace holds
( b1 is satisfying_Ext_Par_Pasch iff for b2, b3, b4, b5, b6 being Element of b1 holds
( Mid b6,b3,b4 & LIN b6,b2,b5 & b2,b3 '||' b4,b5 & not LIN b6,b2,b3 implies Mid b6,b2,b5 ) );

notation
let c1 be OAffinSpace;
synonym c1 satisfies_Ext_Par_Pasch for satisfying_Ext_Par_Pasch c1c1 satisfies_Ext_Par_Paschsatisfies_Ext_Par_Pasch ;
end;

definition
let c1 be OAffinSpace;
attr a1 is satisfying_Gen_Par_Pasch means :: PASCH:def 3
for b1, b2, b3, b4, b5, b6 being Element of a1 holds
( not LIN b1,b2,b4 & b1,b4 '||' b2,b5 & b1,b4 '||' b3,b6 & Mid b1,b2,b3 & LIN b4,b5,b6 implies Mid b4,b5,b6 );
end;

:: deftheorem Def3 defines satisfying_Gen_Par_Pasch PASCH:def 3 :
for b1 being OAffinSpace holds
( b1 is satisfying_Gen_Par_Pasch iff for b2, b3, b4, b5, b6, b7 being Element of b1 holds
( not LIN b2,b3,b5 & b2,b5 '||' b3,b6 & b2,b5 '||' b4,b7 & Mid b2,b3,b4 & LIN b5,b6,b7 implies Mid b5,b6,b7 ) );

notation
let c1 be OAffinSpace;
synonym c1 satisfies_Gen_Par_Pasch for satisfying_Gen_Par_Pasch c1c1 satisfies_Gen_Par_Paschsatisfies_Gen_Par_Pasch ;
end;

definition
let c1 be OAffinSpace;
attr a1 is satisfying_Ext_Bet_Pasch means :: PASCH:def 4
for b1, b2, b3, b4, b5, b6 being Element of a1 holds
not ( Mid b1,b2,b4 & Mid b2,b5,b3 & not LIN b1,b2,b3 & ( for b7 being Element of a1 holds
not ( Mid b1,b7,b3 & Mid b7,b5,b4 ) ) );
end;

:: deftheorem Def4 defines satisfying_Ext_Bet_Pasch PASCH:def 4 :
for b1 being OAffinSpace holds
( b1 is satisfying_Ext_Bet_Pasch iff for b2, b3, b4, b5, b6, b7 being Element of b1 holds
not ( Mid b2,b3,b5 & Mid b3,b6,b4 & not LIN b2,b3,b4 & ( for b8 being Element of b1 holds
not ( Mid b2,b8,b4 & Mid b8,b6,b5 ) ) ) );

notation
let c1 be OAffinSpace;
synonym c1 satisfies_Ext_Bet_Pasch for satisfying_Ext_Bet_Pasch c1c1 satisfies_Ext_Bet_Paschsatisfies_Ext_Bet_Pasch ;
end;

definition
let c1 be OAffinSpace;
attr a1 is satisfying_Int_Bet_Pasch means :: PASCH:def 5
for b1, b2, b3, b4, b5, b6 being Element of a1 holds
not ( Mid b1,b2,b4 & Mid b1,b5,b3 & not LIN b1,b2,b3 & ( for b7 being Element of a1 holds
not ( Mid b2,b7,b3 & Mid b5,b7,b4 ) ) );
end;

:: deftheorem Def5 defines satisfying_Int_Bet_Pasch PASCH:def 5 :
for b1 being OAffinSpace holds
( b1 is satisfying_Int_Bet_Pasch iff for b2, b3, b4, b5, b6, b7 being Element of b1 holds
not ( Mid b2,b3,b5 & Mid b2,b6,b4 & not LIN b2,b3,b4 & ( for b8 being Element of b1 holds
not ( Mid b3,b8,b4 & Mid b6,b8,b5 ) ) ) );

notation
let c1 be OAffinSpace;
synonym c1 satisfies_Int_Bet_Pasch for satisfying_Int_Bet_Pasch c1c1 satisfies_Int_Bet_Paschsatisfies_Int_Bet_Pasch ;
end;

definition
let c1 be OAffinSpace;
attr a1 is Fanoian means :: PASCH:def 6
for b1, b2, b3, b4 being Element of a1 holds
not ( b1,b2 // b3,b4 & b1,b3 // b2,b4 & not LIN b1,b2,b3 & ( for b5 being Element of a1 holds
not ( Mid b1,b5,b4 & Mid b2,b5,b3 ) ) );
end;

:: deftheorem Def6 defines Fanoian PASCH:def 6 :
for b1 being OAffinSpace holds
( b1 is Fanoian iff for b2, b3, b4, b5 being Element of b1 holds
not ( b2,b3 // b4,b5 & b2,b4 // b3,b5 & not LIN b2,b3,b4 & ( for b6 being Element of b1 holds
not ( Mid b2,b6,b5 & Mid b3,b6,b4 ) ) ) );

notation
let c1 be OAffinSpace;
synonym c1 satisfies_Fano for Fanoian c1c1 satisfies_Fanosatisfies_Fano ;
end;

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 b1 being OAffinSpace
for b2, b3, b4, b5 being Element of b1 holds
not ( b2,b3 // b3,b4 & b3 <> b4 & b2 <> b3 & ( for b6 being Element of b1 holds
not ( b5,b3 // b3,b6 & b5,b2 '||' b4,b6 & b4 <> b6 & b3 <> b6 ) ) )
proof end;

theorem Th8: :: PASCH:8
for b1 being OAffinSpace
for b2, b3, b4, b5 being Element of b1 holds
not ( b2,b3 // b2,b4 & b2 <> b4 & b3 <> b2 & ( for b6 being Element of b1 holds
not ( b2,b5 // b2,b6 & b5,b3 '||' b4,b6 & b4 <> b6 ) ) )
proof end;

theorem Th9: :: PASCH:9
for b1 being OAffinSpace
for b2, b3, b4, b5 being Element of b1 holds
not ( b2,b3 '||' b2,b4 & b2 <> b3 & ( for b6 being Element of b1 holds
not ( b2,b5 '||' b2,b6 & b5,b3 '||' b4,b6 ) ) )
proof end;

theorem Th10: :: PASCH:10
canceled;

theorem Th11: :: PASCH:11
for b1 being OAffinSpace
for b2, b3, b4, b5, b6, b7 being Element of b1 holds
( not LIN b2,b3,b4 & LIN b2,b4,b5 & LIN b2,b3,b6 & LIN b2,b3,b7 & b3,b4 '||' b5,b6 & b3,b4 '||' b5,b7 implies b6 = b7 )
proof end;

theorem Th12: :: PASCH:12
for b1 being OAffinSpace
for b2, b3, b4, b5, b6 being Element of b1 holds
( not LIN b2,b3,b4 & b2,b3 '||' b4,b5 & b2,b3 '||' b4,b6 & b2,b4 '||' b3,b5 & b2,b4 '||' b3,b6 implies b5 = b6 )
proof end;

theorem Th13: :: PASCH:13
for b1 being OAffinSpace
for b2, b3, b4, b5, b6 being Element of b1 holds
( not LIN b2,b3,b4 & Mid b3,b2,b5 & LIN b2,b4,b6 & b3,b4 '||' b6,b5 implies Mid b4,b2,b6 )
proof end;

theorem Th14: :: PASCH:14
for b1 being OAffinSpace holds b1 is satisfying_Int_Par_Pasch
proof end;

theorem Th15: :: PASCH:15
for b1 being OAffinSpace
for b2, b3, b4, b5, b6 being Element of b1 holds
( Mid b2,b3,b4 & LIN b2,b5,b6 & b5,b3 '||' b4,b6 & not LIN b2,b5,b3 implies Mid b2,b5,b6 )
proof end;

theorem Th16: :: PASCH:16
for b1 being OAffinSpace holds b1 is satisfying_Ext_Par_Pasch
proof end;

theorem Th17: :: PASCH:17
for b1 being OAffinSpace
for b2, b3, b4, b5, b6, b7 being Element of b1 holds
( not LIN b2,b3,b4 & b2,b4 '||' b3,b5 & b2,b4 '||' b6,b7 & Mid b2,b3,b6 & LIN b4,b5,b7 implies Mid b4,b5,b7 )
proof end;

theorem Th18: :: PASCH:18
for b1 being OAffinSpace holds b1 is satisfying_Gen_Par_Pasch
proof end;

theorem Th19: :: PASCH:19
for b1 being OAffinSpace
for b2, b3, b4, b5, b6 being Element of b1 holds
( not LIN b2,b3,b4 & b3,b2 // b2,b5 & b4,b2 // b2,b6 & b3,b4 '||' b5,b6 implies b3,b4 // b6,b5 )
proof end;

theorem Th20: :: PASCH:20
for b1 being OAffinSpace
for b2, b3, b4, b5, b6 being Element of b1 holds
( not LIN b2,b3,b4 & b2,b3 // b2,b5 & b2,b4 // b2,b6 & b3,b4 '||' b5,b6 implies b3,b4 // b5,b6 )
proof end;

theorem Th21: :: PASCH:21
for b1 being OAffinSpace
for b2, b3, b4, b5 being Element of b1 holds
( not LIN b2,b3,b4 & b2,b3 '||' b4,b5 & b2,b4 '||' b3,b5 implies ( b2,b3 // b4,b5 & b2,b4 // b3,b5 ) )
proof end;

theorem Th22: :: PASCH:22
for b1 being OAffinSpace
for b2, b3, b4, b5, b6 being Element of b1 holds
( Mid b2,b3,b4 & b3,b5 // b4,b6 & b2,b5 // b2,b6 & not LIN b2,b6,b4 & b2 <> b3 implies Mid b2,b5,b6 )
proof end;

theorem Th23: :: PASCH:23
for b1 being OAffinSpace
for b2, b3, b4, b5, b6 being Element of b1 holds
( Mid b2,b3,b4 & b5,b3 // b6,b4 & b2,b5 // b2,b6 & not LIN b2,b4,b6 & b2 <> b5 implies Mid b2,b5,b6 )
proof end;

theorem Th24: :: PASCH:24
for b1 being OAffinSpace
for b2, b3, b4, b5, b6 being Element of b1 holds
not ( not LIN b2,b3,b4 & b2,b4 // b2,b5 & b4,b3 // b5,b6 & LIN b3,b2,b6 & b2 <> b6 & Mid b3,b2,b6 )
proof end;

theorem Th25: :: PASCH:25
for b1 being OAffinSpace
for b2, b3, b4, b5 being Element of b1 holds
not ( b2,b3 // b2,b4 & b3 <> b2 & ( for b6 being Element of b1 holds
not ( b2,b5 // b2,b6 & b3,b5 // b4,b6 ) ) )
proof end;

theorem Th26: :: PASCH:26
for b1 being OAffinSpace
for b2, b3, b4, b5 being Element of b1 holds
not ( Mid b2,b3,b4 & ( for b6 being Element of b1 holds
not ( Mid b2,b6,b5 & b4,b5 // b3,b6 ) ) )
proof end;

theorem Th27: :: PASCH:27
for b1 being OAffinSpace
for b2, b3, b4, b5 being Element of b1 holds
not ( b2 <> b3 & Mid b2,b3,b4 & ( for b6 being Element of b1 holds
not ( Mid b2,b5,b6 & b3,b5 // b4,b6 ) ) )
proof end;

theorem Th28: :: PASCH:28
for b1 being OAffinSpace
for b2, b3, b4, b5 being Element of b1 holds
not ( not LIN b2,b3,b4 & Mid b2,b5,b4 & ( for b6 being Element of b1 holds
not ( Mid b2,b6,b3 & b3,b4 // b6,b5 ) ) )
proof end;

theorem Th29: :: PASCH:29
for b1 being OAffinSpace
for b2, b3, b4 being Element of b1 holds
ex b5 being Element of b1 st
( b2,b5 // b3,b4 & b2,b3 // b5,b4 )
proof end;

theorem Th30: :: PASCH:30
for b1 being OAffinSpace
for b2, b3, b4, b5 being Element of b1 holds
not ( b2,b3 // b4,b5 & not LIN b2,b3,b4 & ( for b6 being Element of b1 holds
not ( Mid b2,b6,b5 & Mid b3,b6,b4 ) ) )
proof end;

theorem Th31: :: PASCH:31
canceled;

theorem Th32: :: PASCH:32
for b1 being OAffinSpace holds b1 is Fanoian
proof end;

theorem Th33: :: PASCH:33
for b1 being OAffinSpace
for b2, b3, b4, b5 being Element of b1 holds
not ( b2,b3 '||' b4,b5 & b2,b4 '||' b3,b5 & not LIN b2,b3,b4 & ( for b6 being Element of b1 holds
not ( LIN b6,b2,b5 & LIN b6,b3,b4 ) ) )
proof end;

theorem Th34: :: PASCH:34
for b1 being OAffinSpace
for b2, b3, b4, b5, b6 being Element of b1 holds
not ( b2,b3 '||' b4,b5 & b2,b4 '||' b3,b5 & not LIN b2,b3,b4 & LIN b6,b2,b5 & LIN b6,b3,b4 & LIN b6,b2,b3 )
proof end;

theorem Th35: :: PASCH:35
for b1 being OAffinSpace
for b2, b3, b4, b5, b6 being Element of b1 holds
not ( Mid b2,b3,b4 & Mid b3,b5,b6 & not LIN b2,b3,b6 & ( for b7 being Element of b1 holds
not ( Mid b2,b7,b6 & Mid b7,b5,b4 ) ) )
proof end;

theorem Th36: :: PASCH:36
for b1 being OAffinSpace holds b1 is satisfying_Ext_Bet_Pasch
proof end;

theorem Th37: :: PASCH:37
for b1 being OAffinSpace
for b2, b3, b4, b5, b6 being Element of b1 holds
not ( Mid b2,b3,b4 & Mid b2,b5,b6 & not LIN b2,b3,b6 & ( for b7 being Element of b1 holds
not ( Mid b3,b7,b6 & Mid b5,b7,b4 ) ) )
proof end;

theorem Th38: :: PASCH:38
for b1 being OAffinSpace holds b1 is satisfying_Int_Bet_Pasch
proof end;

theorem Th39: :: PASCH:39
for b1 being OAffinSpace
for b2, b3, b4, b5, b6, b7 being Element of b1 holds
( Mid b2,b3,b4 & b2,b3 // b5,b6 & not LIN b2,b3,b5 & LIN b5,b6,b7 & b2,b5 // b3,b6 & b2,b5 // b4,b7 implies Mid b5,b6,b7 )
proof end;