let AP be AffinPlane; :: thesis: ( AP is Desarguesian iff AP is satisfying_DES_1 )
hereby :: thesis: ( AP is satisfying_DES_1 implies AP is Desarguesian )
assume A1: AP is Desarguesian ; :: thesis: AP is satisfying_DES_1
thus AP is satisfying_DES_1 :: thesis: verum
proof
let A be Subset of AP; :: according to AFF_2:def 5 :: thesis: for P, C being Subset of AP
for o, a, b, c, a9, b9, c9 being Element of AP st o in A & o in P & o <> a & o <> b & o <> c & a in A & a9 in A & b in P & b9 in P & c in C & c9 in C & A is being_line & P is being_line & C is being_line & A <> P & A <> C & a,b // a9,b9 & a,c // a9,c9 & b,c // b9,c9 & not LIN a,b,c & c <> c9 holds
o in C
let P, C be Subset of AP; :: thesis: for o, a, b, c, a9, b9, c9 being Element of AP st o in A & o in P & o <> a & o <> b & o <> c & a in A & a9 in A & b in P & b9 in P & c in C & c9 in C & A is being_line & P is being_line & C is being_line & A <> P & A <> C & a,b // a9,b9 & a,c // a9,c9 & b,c // b9,c9 & not LIN a,b,c & c <> c9 holds
o in C
let o, a, b, c, a9, b9, c9 be Element of AP; :: thesis: ( o in A & o in P & o <> a & o <> b & o <> c & a in A & a9 in A & b in P & b9 in P & c in C & c9 in C & A is being_line & P is being_line & C is being_line & A <> P & A <> C & a,b // a9,b9 & a,c // a9,c9 & b,c // b9,c9 & not LIN a,b,c & c <> c9 implies o in C )
assume that
A2: o in A and
A3: o in P and
A4: o <> a and
A5: o <> b and
o <> c and
A6: a in A and
A7: a9 in A and
A8: b in P and
A9: b9 in P and
A10: c in C and
A11: c9 in C and
A12: A is being_line and
A13: P is being_line and
A14: C is being_line and
A15: A <> P and
A16: A <> C and
A17: a,b // a9,b9 and
A18: a,c // a9,c9 and
A19: b,c // b9,c9 and
A20: not LIN a,b,c and
A21: c <> c9 ; :: thesis: o in C
set K = Line (o,c9);
assume A22: not o in C ; :: thesis: contradiction
then A23: Line (o,c9) is being_line by A11, AFF_1:24;
A24: a9 <> c9
proof
assume A25: a9 = c9 ; :: thesis: contradiction
then b,c // a9,b9 by A19, AFF_1:4;
then ( a,b // b,c or a9 = b9 ) by A17, AFF_1:5;
then ( b,a // b,c or a9 = b9 ) by AFF_1:4;
then ( LIN b,a,c or a9 = b9 ) by AFF_1:def 1;
hence contradiction by A2, A3, A7, A9, A11, A12, A13, A15, A20, A22, A25, AFF_1:6, AFF_1:18; :: thesis: verum
end;
A26: A <> Line (o,c9) A28: a <> c by A20, AFF_1:7;
A29: o in Line (o,c9) by A11, A22, AFF_1:24;
A30: c9 in Line (o,c9) by A11, A22, AFF_1:24;
not a,c // Line (o,c9) then consider x being Element of AP such that
A32: x in Line (o,c9) and
A33: LIN a,c,x by A23, AFF_1:59;
A34: o <> x A36: b9 <> c9
proof
assume b9 = c9 ; :: thesis: contradiction
then ( a9 = b9 or a,c // a,b ) by A17, A18, AFF_1:5;
then ( a9 = b9 or LIN a,c,b ) by AFF_1:def 1;
then b,c // a,c by A18, A19, A20, A24, AFF_1:5, AFF_1:6;
then c,b // c,a by AFF_1:4;
then LIN c,b,a by AFF_1:def 1;
hence contradiction by A20, AFF_1:6; :: thesis: verum
end;
A37: a,c // a,x by A33, AFF_1:def 1;
then a,x // a9,c9 by A18, A28, AFF_1:5;
then b,x // b9,c9 by A1, A2, A3, A4, A5, A6, A7, A8, A9, A12, A13, A15, A17, A23, A29, A30, A26, A32, A34;
then A38: b,x // b,c by A19, A36, AFF_1:5;
A39: not LIN a,b,x
proof
assume LIN a,b,x ; :: thesis: contradiction
then a,b // a,x by AFF_1:def 1;
then ( a,b // a,c or a = x ) by A37, AFF_1:5;
hence contradiction by A2, A4, A6, A12, A20, A23, A29, A26, A32, AFF_1:18, AFF_1:def 1; :: thesis: verum
end;
LIN a,x,c by A33, AFF_1:6;
then c in Line (o,c9) by A32, A39, A38, AFF_1:14;
hence contradiction by A10, A11, A14, A21, A22, A23, A29, A30, AFF_1:18; :: thesis: verum
end;
end;
assume A40: AP is satisfying_DES_1 ; :: thesis: AP is Desarguesian
let A be Subset of AP; :: according to AFF_2:def 4 :: thesis: for P, C being Subset of AP
for o, a, b, c, a9, b9, c9 being Element of AP st o in A & o in P & o in C & o <> a & o <> b & o <> c & a in A & a9 in A & b in P & b9 in P & c in C & c9 in C & A is being_line & P is being_line & C is being_line & A <> P & A <> C & a,b // a9,b9 & a,c // a9,c9 holds
b,c // b9,c9
let P, C be Subset of AP; :: thesis: for o, a, b, c, a9, b9, c9 being Element of AP st o in A & o in P & o in C & o <> a & o <> b & o <> c & a in A & a9 in A & b in P & b9 in P & c in C & c9 in C & A is being_line & P is being_line & C is being_line & A <> P & A <> C & a,b // a9,b9 & a,c // a9,c9 holds
b,c // b9,c9
let o, a, b, c, a9, b9, c9 be Element of AP; :: thesis: ( o in A & o in P & o in C & o <> a & o <> b & o <> c & a in A & a9 in A & b in P & b9 in P & c in C & c9 in C & A is being_line & P is being_line & C is being_line & A <> P & A <> C & a,b // a9,b9 & a,c // a9,c9 implies b,c // b9,c9 )
assume that
A41: o in A and
A42: o in P and
A43: o in C and
A44: o <> a and
A45: o <> b and
A46: o <> c and
A47: a in A and
A48: a9 in A and
A49: b in P and
A50: b9 in P and
A51: c in C and
A52: c9 in C and
A53: A is being_line and
A54: P is being_line and
A55: C is being_line and
A56: A <> P and
A57: A <> C and
A58: a,b // a9,b9 and
A59: a,c // a9,c9 ; :: thesis: b,c // b9,c9
assume A60: not b,c // b9,c9 ; :: thesis: contradiction
A61: a9 <> b9
proof
A62: a9,c9 // c,a by A59, AFF_1:4;
assume A63: a9 = b9 ; :: thesis: contradiction
then a9 in C by A41, A42, A43, A48, A50, A53, A54, A56, AFF_1:18;
then ( a in C or a9 = c9 ) by A51, A52, A55, A62, AFF_1:48;
hence contradiction by A41, A43, A44, A47, A53, A55, A57, A60, A63, AFF_1:3, AFF_1:18; :: thesis: verum
end;
A64: a9 <> c9
proof
assume a9 = c9 ; :: thesis: contradiction
then A65: a9 in P by A41, A42, A43, A48, A52, A53, A55, A57, AFF_1:18;
a9,b9 // b,a by A58, AFF_1:4;
then a in P by A49, A50, A54, A61, A65, AFF_1:48;
hence contradiction by A41, A42, A44, A47, A53, A54, A56, AFF_1:18; :: thesis: verum
end;
A66: o <> c9
proof
assume A67: o = c9 ; :: thesis: contradiction
a9,c9 // a,c by A59, AFF_1:4;
then c in A by A41, A47, A48, A53, A64, A67, AFF_1:48;
hence contradiction by A41, A43, A46, A51, A53, A55, A57, AFF_1:18; :: thesis: verum
end;
set M = Line (a,c);
set N = Line (b9,c9);
A68: a <> c by A41, A43, A44, A47, A51, A53, A55, A57, AFF_1:18;
then A69: c in Line (a,c) by AFF_1:24;
A70: a <> b by A41, A42, A44, A47, A49, A53, A54, A56, AFF_1:18;
A71: not LIN a9,b9,c9
proof
assume A72: LIN a9,b9,c9 ; :: thesis: contradiction
then a9,b9 // a9,c9 by AFF_1:def 1;
then a9,b9 // a,c by A59, A64, AFF_1:5;
then a,b // a,c by A58, A61, AFF_1:5;
then LIN a,b,c by AFF_1:def 1;
then LIN b,c,a by AFF_1:6;
then b,c // b,a by AFF_1:def 1;
then b,c // a,b by AFF_1:4;
then A73: b,c // a9,b9 by A58, A70, AFF_1:5;
LIN b9,c9,a9 by A72, AFF_1:6;
then b9,c9 // b9,a9 by AFF_1:def 1;
then b9,c9 // a9,b9 by AFF_1:4;
hence contradiction by A60, A61, A73, AFF_1:5; :: thesis: verum
end;
A74: b9 <> c9 by A60, AFF_1:3;
then A75: ( b9 in Line (b9,c9) & c9 in Line (b9,c9) ) by AFF_1:24;
Line (b9,c9) is being_line by A74, AFF_1:24;
then consider D being Subset of AP such that
A76: b in D and
A77: Line (b9,c9) // D by AFF_1:49;
A78: a in Line (a,c) by A68, AFF_1:24;
A79: not Line (a,c) // D
proof
assume Line (a,c) // D ; :: thesis: contradiction
then Line (a,c) // Line (b9,c9) by A77, AFF_1:44;
then a,c // b9,c9 by A78, A69, A75, AFF_1:39;
then a9,c9 // b9,c9 by A59, A68, AFF_1:5;
then c9,a9 // c9,b9 by AFF_1:4;
then LIN c9,a9,b9 by AFF_1:def 1;
hence contradiction by A71, AFF_1:6; :: thesis: verum
end;
A80: Line (a,c) is being_line by A68, AFF_1:24;
D is being_line by A77, AFF_1:36;
then consider x being Element of AP such that
A81: x in Line (a,c) and
A82: x in D by A80, A79, AFF_1:58;
LIN a,c,x by A80, A78, A69, A81, AFF_1:21;
then a,c // a,x by AFF_1:def 1;
then A83: a,x // a9,c9 by A59, A68, AFF_1:5;
set T = Line (c9,x);
A84: a <> a9
proof
assume A85: a = a9 ; :: thesis: contradiction
then LIN a,c,c9 by A59, AFF_1:def 1;
then LIN c,c9,a by AFF_1:6;
then A86: ( c = c9 or a in C ) by A51, A52, A55, AFF_1:25;
LIN a,b,b9 by A58, A85, AFF_1:def 1;
then LIN b,b9,a by AFF_1:6;
then ( b = b9 or a in P ) by A49, A50, A54, AFF_1:25;
hence contradiction by A41, A42, A43, A44, A47, A53, A54, A55, A56, A57, A60, A86, AFF_1:2, AFF_1:18; :: thesis: verum
end;
A87: x <> c9
proof
assume x = c9 ; :: thesis: contradiction
then c9,a // c9,a9 by A83, AFF_1:4;
then LIN c9,a,a9 by AFF_1:def 1;
then LIN a,a9,c9 by AFF_1:6;
then c9 in A by A47, A48, A53, A84, AFF_1:25;
hence contradiction by A41, A43, A52, A53, A55, A57, A66, AFF_1:18; :: thesis: verum
end;
then A88: ( Line (c9,x) is being_line & c9 in Line (c9,x) ) by AFF_1:24;
A89: b,x // b9,c9 by A75, A76, A77, A82, AFF_1:39;
A90: x in Line (c9,x) by A87, AFF_1:24;
A91: a <> x
proof
assume a = x ; :: thesis: contradiction
then a,b // b9,c9 by A75, A76, A77, A82, AFF_1:39;
then a9,b9 // b9,c9 by A58, A70, AFF_1:5;
then b9,a9 // b9,c9 by AFF_1:4;
then LIN b9,a9,c9 by AFF_1:def 1;
hence contradiction by A71, AFF_1:6; :: thesis: verum
end;
not LIN a,b,x
proof
assume LIN a,b,x ; :: thesis: contradiction
then a,b // a,x by AFF_1:def 1;
then a,b // a9,c9 by A83, A91, AFF_1:5;
then a9,b9 // a9,c9 by A58, A70, AFF_1:5;
hence contradiction by A71, AFF_1:def 1; :: thesis: verum
end;
then o in Line (c9,x) by A40, A41, A42, A44, A45, A47, A48, A49, A50, A53, A54, A56, A58, A83, A89, A87, A88, A90;
then x in C by A43, A52, A55, A66, A88, A90, AFF_1:18;
then C = Line (a,c) by A51, A55, A60, A80, A69, A81, A89, AFF_1:18;
hence contradiction by A41, A43, A44, A47, A53, A55, A57, A78, AFF_1:18; :: thesis: verum