let AP be AffinPlane; :: thesis: ( AP is satisfying_TDES_2 implies AP is satisfying_TDES_3 )
assume A1: AP is satisfying_TDES_2 ; :: thesis: AP is satisfying_TDES_3
let K be Subset of AP; :: according to AFF_2:def 10 :: thesis: for o, a, b, c, a9, b9, c9 being Element of AP st K is being_line & o in K & c in K & not a in K & o <> c & a <> b & LIN o,a,a9 & LIN o,b,b9 & a,b // a9,b9 & a,c // a9,c9 & b,c // b9,c9 & a,b // K holds
c9 in K
let o, a, b, c, a9, b9, c9 be Element of AP; :: thesis: ( K is being_line & o in K & c in K & not a in K & o <> c & a <> b & LIN o,a,a9 & LIN o,b,b9 & a,b // a9,b9 & a,c // a9,c9 & b,c // b9,c9 & a,b // K implies c9 in K )
assume that
A2: K is being_line and
A3: o in K and
A4: c in K and
A5: not a in K and
A6: o <> c and
A7: a <> b and
A8: LIN o,a,a9 and
A9: LIN o,b,b9 and
A10: a,b // a9,b9 and
A11: a,c // a9,c9 and
A12: b,c // b9,c9 and
A13: a,b // K ; :: thesis: c9 in K
set A = Line (o,a);
set P = Line (o,b);
set N = Line (b,c);
A14: o in Line (o,a) by A3, A5, AFF_1:24;
A15: not LIN a,b,c
proof
assume LIN a,b,c ; :: thesis: contradiction
then a,b // a,c by AFF_1:def 1;
then a,c // K by A7, A13, AFF_1:32;
then c,a // K by AFF_1:34;
hence contradiction by A2, A4, A5, AFF_1:23; :: thesis: verum
end;
A16: o <> b by A3, A5, A13, AFF_1:35;
then A17: b in Line (o,b) by AFF_1:24;
A18: a9,b9 // b,a by A10, AFF_1:4;
A19: b <> c by A4, A5, A13, AFF_1:35;
then A20: ( b in Line (b,c) & c in Line (b,c) ) by AFF_1:24;
A21: a in Line (o,a) by A3, A5, AFF_1:24;
A22: Line (o,a) is being_line by A3, A5, AFF_1:24;
A23: Line (o,a) <> Line (o,b)
proof
assume Line (o,a) = Line (o,b) ; :: thesis: contradiction
then a,b // Line (o,a) by A22, A21, A17, AFF_1:40, AFF_1:41;
hence contradiction by A3, A5, A7, A13, A14, A21, AFF_1:45, AFF_1:53; :: thesis: verum
end;
assume A24: not c9 in K ; :: thesis: contradiction
A25: Line (o,b) is being_line by A16, AFF_1:24;
A26: o in Line (o,b) by A16, AFF_1:24;
then A27: b9 in Line (o,b) by A9, A16, A25, A17, AFF_1:25;
A28: a9 in Line (o,a) by A3, A5, A8, A22, A14, A21, AFF_1:25;
A29: a9 <> b9
proof
assume A30: a9 = b9 ; :: thesis: contradiction
then ( a,c // b,c or a9 = c9 ) by A11, A12, AFF_1:5;
then ( c,a // c,b or a9 = c9 ) by AFF_1:4;
then ( LIN c,a,b or a9 = c9 ) by AFF_1:def 1;
hence contradiction by A3, A24, A15, A22, A25, A14, A26, A28, A27, A23, A30, AFF_1:6, AFF_1:18; :: thesis: verum
end;
A31: a9 <> c9
proof
assume a9 = c9 ; :: thesis: contradiction
then b,c // a9,b9 by A12, AFF_1:4;
then a,b // b,c by A10, A29, AFF_1:5;
then b,a // b,c by AFF_1:4;
then LIN b,a,c by AFF_1:def 1;
hence contradiction by A15, AFF_1:6; :: thesis: verum
end;
not a9,c9 // K
proof
assume A32: a9,c9 // K ; :: thesis: contradiction
a9,c9 // a,c by A11, AFF_1:4;
then a,c // K by A31, A32, AFF_1:32;
then c,a // K by AFF_1:34;
hence contradiction by A2, A4, A5, AFF_1:23; :: thesis: verum
end;
then consider x being Element of AP such that
A33: x in K and
A34: LIN a9,c9,x by A2, AFF_1:59;
a9,c9 // a9,x by A34, AFF_1:def 1;
then A35: a,c // a9,x by A11, A31, AFF_1:5;
Line (b,c) is being_line by A19, AFF_1:24;
then consider T being Subset of AP such that
A36: x in T and
A37: Line (b,c) // T by AFF_1:49;
A38: not b in K by A5, A13, AFF_1:35;
A39: not T // Line (o,b)
proof
assume T // Line (o,b) ; :: thesis: contradiction
then Line (b,c) // Line (o,b) by A37, AFF_1:44;
then c in Line (o,b) by A17, A20, AFF_1:45;
hence contradiction by A2, A3, A4, A6, A38, A25, A26, A17, AFF_1:18; :: thesis: verum
end;
T is being_line by A37, AFF_1:36;
then consider y being Element of AP such that
A40: y in T and
A41: y in Line (o,b) by A25, A39, AFF_1:58;
A42: b,c // y,x by A20, A36, A37, A40, AFF_1:39;
A43: now :: thesis: not y = b9
assume y = b9 ; :: thesis: contradiction
then b9,c9 // b9,x by A12, A19, A42, AFF_1:5;
then LIN b9,c9,x by AFF_1:def 1;
then A44: LIN c9,x,b9 by AFF_1:6;
( LIN c9,x,a9 & LIN c9,x,c9 ) by A34, AFF_1:6, AFF_1:7;
then LIN a9,b9,c9 by A24, A33, A44, AFF_1:8;
then a9,b9 // a9,c9 by AFF_1:def 1;
then a9,b9 // a,c by A11, A31, AFF_1:5;
then a,b // a,c by A10, A29, AFF_1:5;
hence contradiction by A15, AFF_1:def 1; :: thesis: verum
end;
LIN o,b,y by A25, A26, A17, A41, AFF_1:21;
then a,b // a9,y by A1, A2, A3, A4, A5, A6, A7, A8, A13, A33, A42, A35;
then a9,b9 // a9,y by A7, A10, AFF_1:5;
then LIN a9,b9,y by AFF_1:def 1;
then LIN b9,y,a9 by AFF_1:6;
then a9 in Line (o,b) by A25, A27, A41, A43, AFF_1:25;
then a in Line (o,b) by A25, A17, A27, A29, A18, AFF_1:48;
hence contradiction by A3, A5, A25, A26, A23, AFF_1:24; :: thesis: verum