let AP be AffinPlane; ( AP is Desarguesian implies AP is satisfying_DES1 )
assume A1:
AP is Desarguesian
; AP is satisfying_DES1
let A be Subset of AP; AFF_3:def 1 for P, C being Subset of AP
for o, a, a9, b, b9, c, c9, p, q being Element of AP st A is being_line & P is being_line & C is being_line & P <> A & P <> C & A <> C & o in A & a in A & a9 in A & o in P & b in P & b9 in P & o in C & c in C & c9 in C & o <> a & o <> b & o <> c & p <> q & not LIN b,a,c & not LIN b9,a9,c9 & a <> a9 & LIN b,a,p & LIN b9,a9,p & LIN b,c,q & LIN b9,c9,q & a,c // a9,c9 holds
a,c // p,q
let P, C be Subset of AP; for o, a, a9, b, b9, c, c9, p, q being Element of AP st A is being_line & P is being_line & C is being_line & P <> A & P <> C & A <> C & o in A & a in A & a9 in A & o in P & b in P & b9 in P & o in C & c in C & c9 in C & o <> a & o <> b & o <> c & p <> q & not LIN b,a,c & not LIN b9,a9,c9 & a <> a9 & LIN b,a,p & LIN b9,a9,p & LIN b,c,q & LIN b9,c9,q & a,c // a9,c9 holds
a,c // p,q
let o, a, a9, b, b9, c, c9, p, q be Element of AP; ( A is being_line & P is being_line & C is being_line & P <> A & P <> C & A <> C & o in A & a in A & a9 in A & o in P & b in P & b9 in P & o in C & c in C & c9 in C & o <> a & o <> b & o <> c & p <> q & not LIN b,a,c & not LIN b9,a9,c9 & a <> a9 & LIN b,a,p & LIN b9,a9,p & LIN b,c,q & LIN b9,c9,q & a,c // a9,c9 implies a,c // p,q )
assume that
A2:
A is being_line
and
A3:
P is being_line
and
A4:
C is being_line
and
A5:
P <> A
and
A6:
P <> C
and
A7:
A <> C
and
A8:
o in A
and
A9:
a in A
and
A10:
a9 in A
and
A11:
o in P
and
A12:
b in P
and
A13:
b9 in P
and
A14:
o in C
and
A15:
c in C
and
A16:
c9 in C
and
A17:
o <> a
and
A18:
o <> b
and
A19:
o <> c
and
p <> q
and
A20:
not LIN b,a,c
and
A21:
not LIN b9,a9,c9
and
A22:
a <> a9
and
A23:
LIN b,a,p
and
A24:
LIN b9,a9,p
and
A25:
LIN b,c,q
and
A26:
LIN b9,c9,q
and
A27:
a,c // a9,c9
; a,c // p,q
set D = Line (b,c);
b <> c
by A20, AFF_1:7;
then
Line (b,c) is being_line
by AFF_1:def 3;
then consider D9 being Subset of AP such that
A28:
c9 in D9
and
A29:
Line (b,c) // D9
by AFF_1:49;
A30:
D9 is being_line
by A29, AFF_1:36;
set M = Line (b9,c9);
A31:
q in Line (b9,c9)
by A26, AFF_1:def 2;
A32:
b in Line (b,c)
by AFF_1:15;
A33:
c in Line (b,c)
by AFF_1:15;
not D9 // P
proof
assume
D9 // P
;
contradiction
then
Line (
b,
c)
// P
by A29, AFF_1:44;
then
c in P
by A12, A32, A33, AFF_1:45;
hence
contradiction
by A3, A4, A6, A11, A14, A15, A19, AFF_1:18;
verum
end;
then consider d being Element of AP such that
A34:
d in D9
and
A35:
d in P
by A3, A30, AFF_1:58;
A36:
q in Line (b,c)
by A25, AFF_1:def 2;
then A37:
d,c9 // b,q
by A32, A28, A29, A34, AFF_1:39;
A38:
( a <> b & b,a // b,p )
by A20, A23, AFF_1:7, AFF_1:def 1;
A39:
c,a // c9,a9
by A27, AFF_1:4;
c,b // c9,d
by A32, A33, A28, A29, A34, AFF_1:39;
then
b,a // d,a9
by A1, A2, A3, A4, A6, A7, A8, A9, A10, A11, A12, A14, A15, A16, A17, A18, A19, A35, A39;
then A40:
d,a9 // b,p
by A38, AFF_1:5;
set K = Line (b9,a9);
A41:
( b9 in Line (b9,a9) & p in Line (b9,a9) )
by A24, AFF_1:15, AFF_1:def 2;
A42:
a9 <> b9
by A21, AFF_1:7;
then A43:
Line (b9,a9) is being_line
by AFF_1:def 3;
A44:
b9 in Line (b9,c9)
by AFF_1:15;
A45:
c9 in Line (b9,c9)
by AFF_1:15;
A46:
b9 <> c9
by A21, AFF_1:7;
then A47:
Line (b9,c9) is being_line
by AFF_1:def 3;
A48:
not LIN o,a,c
proof
assume
LIN o,
a,
c
;
contradiction
then
c in A
by A2, A8, A9, A17, AFF_1:25;
hence
contradiction
by A2, A4, A7, A8, A14, A15, A19, AFF_1:18;
verum
end;
A49:
c <> c9
proof
assume
c = c9
;
contradiction
then A50:
c,
a // c,
a9
by A27, AFF_1:4;
(
LIN o,
a,
a9 & not
LIN o,
c,
a )
by A2, A8, A9, A10, A48, AFF_1:6, AFF_1:21;
hence
contradiction
by A22, A50, AFF_1:14;
verum
end;
A51:
d <> b9
proof
assume
d = b9
;
contradiction
then
Line (
b9,
c9)
= D9
by A46, A47, A44, A45, A28, A30, A34, AFF_1:18;
then
Line (
b,
c)
= Line (
b9,
c9)
by A31, A36, A29, AFF_1:45;
then
LIN c,
c9,
b
by A47, A45, A32, A33, AFF_1:21;
then
b in C
by A4, A15, A16, A49, AFF_1:25;
hence
contradiction
by A3, A4, A6, A11, A12, A14, A18, AFF_1:18;
verum
end;
A52:
a9 <> c9
by A21, AFF_1:7;
A53:
o <> a9
by A4, A14, A15, A16, A27, A52, A48, AFF_1:21, AFF_1:55;
o <> c9
proof
A54:
not
LIN o,
c,
a
by A48, AFF_1:6;
assume A55:
o = c9
;
contradiction
(
LIN o,
a,
a9 &
c,
a // c9,
a9 )
by A2, A8, A9, A10, A27, AFF_1:4, AFF_1:21;
hence
contradiction
by A53, A55, A54, AFF_1:55;
verum
end;
then A56:
Line (b9,c9) <> P
by A3, A4, A6, A11, A14, A16, A45, AFF_1:18;
A57:
a9 in Line (b9,a9)
by AFF_1:15;
then
Line (b9,a9) <> P
by A2, A3, A5, A8, A10, A11, A53, AFF_1:18;
then
a9,c9 // p,q
by A1, A3, A12, A13, A42, A46, A43, A47, A57, A41, A44, A45, A31, A56, A35, A51, A40, A37;
hence
a,c // p,q
by A27, A52, AFF_1:5; verum