:: AFF_1 semantic presentation

definition
let c1 be AffinSpace;
let c2, c3, c4 be Element of c1;
pred LIN c2,c3,c4 means :Def1: :: AFF_1:def 1
a2,a3 // a2,a4;
end;

:: deftheorem Def1 defines LIN AFF_1:def 1 :
for b1 being AffinSpace
for b2, b3, b4 being Element of b1 holds
( LIN b2,b3,b4 iff b2,b3 // b2,b4 );

theorem Th1: :: AFF_1:1
canceled;

theorem Th2: :: AFF_1:2
canceled;

theorem Th3: :: AFF_1:3
canceled;

theorem Th4: :: AFF_1:4
canceled;

theorem Th5: :: AFF_1:5
canceled;

theorem Th6: :: AFF_1:6
canceled;

theorem Th7: :: AFF_1:7
canceled;

theorem Th8: :: AFF_1:8
canceled;

theorem Th9: :: AFF_1:9
canceled;

theorem Th10: :: AFF_1:10
for b1 being AffinSpace
for b2 being Element of b1 holds
not for b3 being Element of b1 holds not b2 <> b3
proof end;

theorem Th11: :: AFF_1:11
for b1 being AffinSpace
for b2, b3 being Element of b1 holds
( b2,b3 // b3,b2 & b2,b3 // b2,b3 )
proof end;

Lemma4: for b1 being AffinSpace
for b2, b3, b4, b5 being Element of b1 holds
( b2,b3 // b4,b5 implies b4,b5 // b2,b3 )
proof end;

theorem Th12: :: AFF_1:12
for b1 being AffinSpace
for b2, b3, b4 being Element of b1 holds
( b2,b3 // b4,b4 & b4,b4 // b2,b3 )
proof end;

Lemma6: for b1 being AffinSpace
for b2, b3, b4, b5 being Element of b1 holds
( b2,b3 // b4,b5 implies b3,b2 // b4,b5 )
proof end;

Lemma7: for b1 being AffinSpace
for b2, b3, b4, b5 being Element of b1 holds
( b2,b3 // b4,b5 implies b2,b3 // b5,b4 )
proof end;

theorem Th13: :: AFF_1:13
for b1 being AffinSpace
for b2, b3, b4, b5 being Element of b1 holds
( b2,b3 // b4,b5 implies ( b2,b3 // b5,b4 & b3,b2 // b4,b5 & b3,b2 // b5,b4 & b4,b5 // b2,b3 & b4,b5 // b3,b2 & b5,b4 // b2,b3 & b5,b4 // b3,b2 ) )
proof end;

theorem Th14: :: AFF_1:14
for b1 being AffinSpace
for b2, b3, b4, b5, b6, b7 being Element of b1 holds
( b2 <> b3 & not ( not ( b2,b3 // b4,b5 & b2,b3 // b6,b7 ) & not ( b2,b3 // b4,b5 & b6,b7 // b2,b3 ) & not ( b4,b5 // b2,b3 & b6,b7 // b2,b3 ) & not ( b4,b5 // b2,b3 & b2,b3 // b6,b7 ) ) implies b4,b5 // b6,b7 )
proof end;

Lemma10: for b1 being AffinSpace
for b2, b3, b4 being Element of b1 holds
( LIN b2,b3,b4 implies ( LIN b2,b4,b3 & LIN b3,b2,b4 ) )
proof end;

theorem Th15: :: AFF_1:15
for b1 being AffinSpace
for b2, b3, b4 being Element of b1 holds
( LIN b2,b3,b4 implies ( LIN b2,b4,b3 & LIN b3,b2,b4 & LIN b3,b4,b2 & LIN b4,b2,b3 & LIN b4,b3,b2 ) )
proof end;

theorem Th16: :: AFF_1:16
for b1 being AffinSpace
for b2, b3 being Element of b1 holds
( LIN b2,b2,b3 & LIN b2,b3,b3 & LIN b2,b3,b2 )
proof end;

theorem Th17: :: AFF_1:17
for b1 being AffinSpace
for b2, b3, b4, b5, b6 being Element of b1 holds
( b2 <> b3 & LIN b2,b3,b4 & LIN b2,b3,b5 & LIN b2,b3,b6 implies LIN b4,b5,b6 )
proof end;

theorem Th18: :: AFF_1:18
for b1 being AffinSpace
for b2, b3, b4, b5 being Element of b1 holds
( b2 <> b3 & LIN b2,b3,b4 & b2,b3 // b4,b5 implies LIN b2,b3,b5 )
proof end;

theorem Th19: :: AFF_1:19
for b1 being AffinSpace
for b2, b3, b4, b5 being Element of b1 holds
( LIN b2,b3,b4 & LIN b2,b3,b5 implies b2,b3 // b4,b5 )
proof end;

theorem Th20: :: AFF_1:20
for b1 being AffinSpace
for b2, b3, b4, b5, b6 being Element of b1 holds
( b2 <> b3 & LIN b4,b5,b2 & LIN b4,b5,b3 & LIN b2,b3,b6 implies LIN b4,b5,b6 )
proof end;

theorem Th21: :: AFF_1:21
for b1 being AffinSpace holds
not for b2, b3, b4 being Element of b1 holds LIN b2,b3,b4
proof end;

theorem Th22: :: AFF_1:22
for b1 being AffinSpace
for b2, b3 being Element of b1 holds
not ( b2 <> b3 & ( for b4 being Element of b1 holds LIN b2,b3,b4 ) )
proof end;

theorem Th23: :: AFF_1:23
for b1 being AffinSpace
for b2, b3, b4, b5 being Element of b1 holds
( not LIN b2,b3,b4 & LIN b2,b4,b5 & b3,b4 // b3,b5 implies b4 = b5 )
proof end;

definition
let c1 be AffinSpace;
let c2, c3 be Element of c1;
func Line c2,c3 -> Subset of a1 means :Def2: :: AFF_1:def 2
for b1 being Element of a1 holds
( b1 in a4 iff LIN a2,a3,b1 );
existence
ex b1 being Subset of c1 st
for b2 being Element of c1 holds
( b2 in b1 iff LIN c2,c3,b2 )
proof end;
uniqueness
for b1, b2 being Subset of c1 holds
( ( for b3 being Element of c1 holds
( b3 in b1 iff LIN c2,c3,b3 ) ) & ( for b3 being Element of c1 holds
( b3 in b2 iff LIN c2,c3,b3 ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def2 defines Line AFF_1:def 2 :
for b1 being AffinSpace
for b2, b3 being Element of b1
for b4 being Subset of b1 holds
( b4 = Line b2,b3 iff for b5 being Element of b1 holds
( b5 in b4 iff LIN b2,b3,b5 ) );

Lemma19: for b1 being AffinSpace
for b2, b3 being Element of b1 holds Line b2,b3 c= Line b3,b2
proof end;

theorem Th24: :: AFF_1:24
canceled;

theorem Th25: :: AFF_1:25
for b1 being AffinSpace
for b2, b3 being Element of b1 holds Line b2,b3 = Line b3,b2
proof end;

definition
let c1 be AffinSpace;
let c2, c3 be Element of c1;
redefine func Line as Line c2,c3 -> Subset of a1;
commutativity
for b1, b2 being Element of c1 holds Line b1,b2 = Line b2,b1
by Th25;
end;

theorem Th26: :: AFF_1:26
for b1 being AffinSpace
for b2, b3 being Element of b1 holds
( b2 in Line b2,b3 & b3 in Line b2,b3 )
proof end;

theorem Th27: :: AFF_1:27
for b1 being AffinSpace
for b2, b3, b4, b5 being Element of b1 holds
( b2 in Line b3,b4 & b5 in Line b3,b4 & b2 <> b5 implies Line b2,b5 c= Line b3,b4 )
proof end;

theorem Th28: :: AFF_1:28
for b1 being AffinSpace
for b2, b3, b4, b5 being Element of b1 holds
( b2 in Line b3,b4 & b5 in Line b3,b4 & b3 <> b4 implies Line b3,b4 c= Line b2,b5 )
proof end;

definition
let c1 be AffinSpace;
let c2 be Subset of c1;
attr a2 is being_line means :Def3: :: AFF_1:def 3
ex b1, b2 being Element of a1 st
( b1 <> b2 & a2 = Line b1,b2 );
end;

:: deftheorem Def3 defines being_line AFF_1:def 3 :
for b1 being AffinSpace
for b2 being Subset of b1 holds
( b2 is being_line iff ex b3, b4 being Element of b1 st
( b3 <> b4 & b2 = Line b3,b4 ) );

notation
let c1 be AffinSpace;
let c2 be Subset of c1;
synonym c2 is_line for being_line c2;
end;

Lemma25: for b1 being AffinSpace
for b2, b3 being Element of b1
for b4 being Subset of b1 holds
( b4 is_line & b2 in b4 & b3 in b4 & b2 <> b3 implies b4 = Line b2,b3 )
proof end;

theorem Th29: :: AFF_1:29
canceled;

theorem Th30: :: AFF_1:30
for b1 being AffinSpace
for b2, b3 being Element of b1
for b4, b5 being Subset of b1 holds
not ( b4 is_line & b5 is_line & b2 in b4 & b3 in b4 & b2 in b5 & b3 in b5 & not b2 = b3 & not b4 = b5 )
proof end;

theorem Th31: :: AFF_1:31
for b1 being AffinSpace
for b2 being Subset of b1 holds
not ( b2 is_line & ( for b3, b4 being Element of b1 holds
not ( b3 in b2 & b4 in b2 & b3 <> b4 ) ) )
proof end;

theorem Th32: :: AFF_1:32
for b1 being AffinSpace
for b2 being Element of b1
for b3 being Subset of b1 holds
not ( b3 is_line & ( for b4 being Element of b1 holds
not ( b2 <> b4 & b4 in b3 ) ) )
proof end;

theorem Th33: :: AFF_1:33
for b1 being AffinSpace
for b2, b3, b4 being Element of b1 holds
( LIN b2,b3,b4 iff ex b5 being Subset of b1 st
( b5 is_line & b2 in b5 & b3 in b5 & b4 in b5 ) )
proof end;

definition
let c1 be AffinSpace;
let c2, c3 be Element of c1;
let c4 be Subset of c1;
pred c2,c3 // c4 means :Def4: :: AFF_1:def 4
ex b1, b2 being Element of a1 st
( b1 <> b2 & a4 = Line b1,b2 & a2,a3 // b1,b2 );
end;

:: deftheorem Def4 defines // AFF_1:def 4 :
for b1 being AffinSpace
for b2, b3 being Element of b1
for b4 being Subset of b1 holds
( b2,b3 // b4 iff ex b5, b6 being Element of b1 st
( b5 <> b6 & b4 = Line b5,b6 & b2,b3 // b5,b6 ) );

definition
let c1 be AffinSpace;
let c2, c3 be Subset of c1;
pred c2 // c3 means :Def5: :: AFF_1:def 5
ex b1, b2 being Element of a1 st
( a2 = Line b1,b2 & b1 <> b2 & b1,b2 // a3 );
end;

:: deftheorem Def5 defines // AFF_1:def 5 :
for b1 being AffinSpace
for b2, b3 being Subset of b1 holds
( b2 // b3 iff ex b4, b5 being Element of b1 st
( b2 = Line b4,b5 & b4 <> b5 & b4,b5 // b3 ) );

theorem Th34: :: AFF_1:34
canceled;

theorem Th35: :: AFF_1:35
canceled;

theorem Th36: :: AFF_1:36
for b1 being AffinSpace
for b2, b3, b4, b5 being Element of b1 holds
( b2 in Line b3,b4 & b3 <> b4 implies ( b5 in Line b3,b4 iff b3,b4 // b2,b5 ) )
proof end;

theorem Th37: :: AFF_1:37
for b1 being AffinSpace
for b2, b3 being Element of b1
for b4 being Subset of b1 holds
( b4 is_line & b2 in b4 implies ( b3 in b4 iff b2,b3 // b4 ) )
proof end;

theorem Th38: :: AFF_1:38
for b1 being AffinSpace
for b2, b3 being Element of b1
for b4 being Subset of b1 holds
( ( b2 <> b3 & b4 = Line b2,b3 implies ( b4 is_line & b2 in b4 & b3 in b4 & b2 <> b3 ) ) & ( b4 is_line & b2 in b4 & b3 in b4 & b2 <> b3 implies ( b2 <> b3 & b4 = Line b2,b3 ) ) ) by Def3, Lemma25, Th26;

theorem Th39: :: AFF_1:39
for b1 being AffinSpace
for b2, b3, b4 being Element of b1
for b5 being Subset of b1 holds
( b5 is_line & b2 in b5 & b3 in b5 & b2 <> b3 & LIN b2,b3,b4 implies b4 in b5 )
proof end;

theorem Th40: :: AFF_1:40
for b1 being AffinSpace
for b2 being Subset of b1 holds
( ex b3, b4 being Element of b1 st b3,b4 // b2 implies b2 is_line )
proof end;

theorem Th41: :: AFF_1:41
for b1 being AffinSpace
for b2, b3, b4, b5 being Element of b1
for b6 being Subset of b1 holds
( b2 in b6 & b3 in b6 & b6 is_line & b2 <> b3 implies ( b4,b5 // b6 iff b4,b5 // b2,b3 ) )
proof end;

theorem Th42: :: AFF_1:42
canceled;

theorem Th43: :: AFF_1:43
for b1 being AffinSpace
for b2, b3 being Element of b1 holds
( b2 <> b3 implies b2,b3 // Line b2,b3 )
proof end;

theorem Th44: :: AFF_1:44
for b1 being AffinSpace
for b2, b3 being Element of b1
for b4 being Subset of b1 holds
( b4 is_line implies ( b2,b3 // b4 iff ex b5, b6 being Element of b1 st
( b5 <> b6 & b5 in b4 & b6 in b4 & b2,b3 // b5,b6 ) ) )
proof end;

theorem Th45: :: AFF_1:45
for b1 being AffinSpace
for b2, b3, b4, b5 being Element of b1
for b6 being Subset of b1 holds
( b6 is_line & b2,b3 // b6 & b4,b5 // b6 implies b2,b3 // b4,b5 )
proof end;

theorem Th46: :: AFF_1:46
for b1 being AffinSpace
for b2, b3, b4, b5 being Element of b1
for b6 being Subset of b1 holds
( b2,b3 // b6 & b2,b3 // b4,b5 & b2 <> b3 implies b4,b5 // b6 )
proof end;

theorem Th47: :: AFF_1:47
for b1 being AffinSpace
for b2 being Element of b1
for b3 being Subset of b1 holds
( b3 is_line implies b2,b2 // b3 )
proof end;

theorem Th48: :: AFF_1:48
for b1 being AffinSpace
for b2, b3 being Element of b1
for b4 being Subset of b1 holds
( b2,b3 // b4 implies b3,b2 // b4 )
proof end;

theorem Th49: :: AFF_1:49
for b1 being AffinSpace
for b2, b3 being Element of b1
for b4 being Subset of b1 holds
not ( b2,b3 // b4 & not b2 in b4 & b3 in b4 )
proof end;

theorem Th50: :: AFF_1:50
for b1 being AffinSpace
for b2, b3 being Subset of b1 holds
( b2 // b3 implies ( b2 is_line & b3 is_line ) )
proof end;

theorem Th51: :: AFF_1:51
for b1 being AffinSpace
for b2, b3 being Subset of b1 holds
( b2 // b3 iff ex b4, b5, b6, b7 being Element of b1 st
( b4 <> b5 & b6 <> b7 & b4,b5 // b6,b7 & b2 = Line b4,b5 & b3 = Line b6,b7 ) )
proof end;

theorem Th52: :: AFF_1:52
for b1 being AffinSpace
for b2, b3, b4, b5 being Element of b1
for b6, b7 being Subset of b1 holds
( b6 is_line & b7 is_line & b2 in b6 & b3 in b6 & b4 in b7 & b5 in b7 & b2 <> b3 & b4 <> b5 implies ( b6 // b7 iff b2,b3 // b4,b5 ) )
proof end;

theorem Th53: :: AFF_1:53
for b1 being AffinSpace
for b2, b3, b4, b5 being Element of b1
for b6, b7 being Subset of b1 holds
( b2 in b6 & b3 in b6 & b4 in b7 & b5 in b7 & b6 // b7 implies b2,b3 // b4,b5 )
proof end;

theorem Th54: :: AFF_1:54
for b1 being AffinSpace
for b2, b3 being Element of b1
for b4, b5 being Subset of b1 holds
( b2 in b4 & b3 in b4 & b4 // b5 implies b2,b3 // b5 )
proof end;

theorem Th55: :: AFF_1:55
for b1 being AffinSpace
for b2 being Subset of b1 holds
( b2 is_line implies b2 // b2 )
proof end;

theorem Th56: :: AFF_1:56
for b1 being AffinSpace
for b2, b3 being Subset of b1 holds
( b2 // b3 implies b3 // b2 )
proof end;

definition
let c1 be AffinSpace;
let c2, c3 be Subset of c1;
redefine pred // as c2 // c3;
symmetry
for b1, b2 being Subset of c1 holds
( b1 // b2 implies b2 // b1 )
by Th56;
end;

theorem Th57: :: AFF_1:57
for b1 being AffinSpace
for b2, b3 being Element of b1
for b4, b5 being Subset of b1 holds
( b2,b3 // b4 & b4 // b5 implies b2,b3 // b5 )
proof end;

Lemma49: for b1 being AffinSpace
for b2, b3, b4 being Subset of b1 holds
( b2 // b3 & b3 // b4 implies b2 // b4 )
proof end;

theorem Th58: :: AFF_1:58
for b1 being AffinSpace
for b2, b3, b4 being Subset of b1 holds
( not ( not ( b2 // b3 & b3 // b4 ) & not ( b2 // b3 & b4 // b3 ) & not ( b3 // b2 & b3 // b4 ) & not ( b3 // b2 & b4 // b3 ) ) implies b2 // b4 ) by Lemma49;

theorem Th59: :: AFF_1:59
for b1 being AffinSpace
for b2 being Element of b1
for b3, b4 being Subset of b1 holds
( b3 // b4 & b2 in b3 & b2 in b4 implies b3 = b4 )
proof end;

theorem Th60: :: AFF_1:60
for b1 being AffinSpace
for b2, b3, b4 being Element of b1
for b5 being Subset of b1 holds
not ( b2 in b5 & not b3 in b5 & b3,b4 // b5 & not b3 = b4 & LIN b2,b3,b4 )
proof end;

theorem Th61: :: AFF_1:61
for b1 being AffinSpace
for b2, b3, b4, b5, b6 being Element of b1
for b7 being Subset of b1 holds
( b2,b3 // b7 & b4,b5 // b7 & LIN b6,b2,b4 & LIN b6,b3,b5 & b6 in b7 & not b2 in b7 & b2 = b3 implies b4 = b5 )
proof end;

theorem Th62: :: AFF_1:62
for b1 being AffinSpace
for b2, b3, b4, b5 being Element of b1
for b6 being Subset of b1 holds
( b6 is_line & b2 in b6 & b3 in b6 & b4 in b6 & b2 <> b3 & b2,b3 // b4,b5 implies b5 in b6 )
proof end;

theorem Th63: :: AFF_1:63
for b1 being AffinSpace
for b2 being Element of b1
for b3 being Subset of b1 holds
not ( b3 is_line & ( for b4 being Subset of b1 holds
not ( b2 in b4 & b3 // b4 ) ) )
proof end;

theorem Th64: :: AFF_1:64
for b1 being AffinSpace
for b2 being Element of b1
for b3, b4, b5 being Subset of b1 holds
( b3 // b4 & b3 // b5 & b2 in b4 & b2 in b5 implies b4 = b5 )
proof end;

theorem Th65: :: AFF_1:65
for b1 being AffinSpace
for b2, b3, b4, b5 being Element of b1
for b6 being Subset of b1 holds
( b6 is_line & b2 in b6 & b3 in b6 & b4 in b6 & b5 in b6 implies b2,b3 // b4,b5 )
proof end;

theorem Th66: :: AFF_1:66
for b1 being AffinSpace
for b2, b3 being Element of b1
for b4 being Subset of b1 holds
( b4 is_line & b2 in b4 & b3 in b4 implies b2,b3 // b4 ) by Th37;

theorem Th67: :: AFF_1:67
for b1 being AffinSpace
for b2, b3 being Element of b1
for b4, b5 being Subset of b1 holds
( b2,b3 // b4 & b2,b3 // b5 & b2 <> b3 implies b4 // b5 )
proof end;

theorem Th68: :: AFF_1:68
for b1 being AffinSpace
for b2, b3, b4, b5, b6 being Element of b1 holds
( not LIN b2,b3,b4 & LIN b2,b3,b5 & LIN b2,b4,b6 & b3,b4 // b5,b6 & b5 = b6 implies ( b5 = b2 & b6 = b2 ) )
proof end;

theorem Th69: :: AFF_1:69
for b1 being AffinSpace
for b2, b3, b4, b5, b6 being Element of b1 holds
( not LIN b2,b3,b4 & LIN b2,b3,b5 & LIN b2,b4,b6 & b3,b4 // b5,b6 & b5 = b2 implies b6 = b2 )
proof end;

theorem Th70: :: AFF_1:70
for b1 being AffinSpace
for b2, b3, b4, b5, b6, b7 being Element of b1 holds
( not LIN b2,b3,b4 & LIN b2,b3,b5 & LIN b2,b4,b6 & LIN b2,b4,b7 & b3,b4 // b5,b6 & b3,b4 // b5,b7 implies b6 = b7 )
proof end;

theorem Th71: :: AFF_1:71
for b1 being AffinSpace
for b2, b3 being Element of b1
for b4 being Subset of b1 holds
( b4 is_line & b2 in b4 & b3 in b4 & b2 <> b3 implies b4 = Line b2,b3 ) by Lemma25;

theorem Th72: :: AFF_1:72
for b1 being AffinPlane
for b2, b3 being Subset of b1 holds
not ( b2 is_line & b3 is_line & not b2 // b3 & ( for b4 being Element of b1 holds
not ( b4 in b2 & b4 in b3 ) ) )
proof end;

theorem Th73: :: AFF_1:73
for b1 being AffinPlane
for b2, b3 being Element of b1
for b4 being Subset of b1 holds
not ( b4 is_line & not b2,b3 // b4 & ( for b5 being Element of b1 holds
not ( b5 in b4 & LIN b2,b3,b5 ) ) )
proof end;

theorem Th74: :: AFF_1:74
for b1 being AffinPlane
for b2, b3, b4, b5 being Element of b1 holds
not ( not b2,b3 // b4,b5 & ( for b6 being Element of b1 holds
not ( LIN b2,b3,b6 & LIN b4,b5,b6 ) ) )
proof end;