let AS be AffinSpace; :: thesis: for a, a9, b, b9, o, x being Element of AS st not LIN o,a,b & LIN o,a,a9 & LIN o,b,b9 & LIN o,b,x & a,b // a9,b9 & a,b // a9,x holds
b9 = x
let a, a9, b, b9, o, x be Element of AS; :: thesis: ( not LIN o,a,b & LIN o,a,a9 & LIN o,b,b9 & LIN o,b,x & a,b // a9,b9 & a,b // a9,x implies b9 = x )
assume that
A1: not LIN o,a,b and
A2: LIN o,a,a9 and
A3: LIN o,b,b9 and
A4: LIN o,b,x and
A5: a,b // a9,b9 and
A6: a,b // a9,x ; :: thesis: b9 = x
set A = Line (o,a);
set C = Line (o,b);
set P = Line (a9,b9);
A7: a9 in Line (a9,b9) by Th14;
assume A8: b9 <> x ; :: thesis: contradiction
A9: a9 <> b9
proof
assume A10: a9 = b9 ; :: thesis: contradiction
then a9 = o by A1, A2, A3, Th53;
hence contradiction by A1, A4, A6, A8, A10, Th54; :: thesis: verum
end;
then A11: Line (a9,b9) is being_line ;
A12: o <> b by A1, Th6;
then A13: Line (o,b) is being_line ;
A14: b9 in Line (a9,b9) by Th14;
a <> b by A1, Th6;
then a9,b9 // a9,x by A5, A6, Th4;
then LIN a9,b9,x ;
then A15: x in Line (a9,b9) by A9, A11, A7, A14, Th24;
A16: b in Line (o,b) by Th14;
A17: o in Line (o,b) by Th14;
then A18: x in Line (o,b) by A4, A12, A13, A16, Th24;
b9 in Line (o,b) by A3, A12, A13, A17, A16, Th24;
then A19: a9 in Line (o,b) by A8, A13, A11, A7, A14, A18, A15, Th17;
A20: o <> a by A1, Th6;
then A21: Line (o,a) is being_line ;
A22: a9 <> o
proof
assume A23: a9 = o ; :: thesis: contradiction
then b9 = o by A1, A3, A5, Th54;
hence contradiction by A1, A4, A6, A8, A23, Th54; :: thesis: verum
end;
A24: o in Line (o,a) by Th14;
A25: a in Line (o,a) by Th14;
then a9 in Line (o,a) by A2, A20, A21, A24, Th24;
then b in Line (o,a) by A22, A21, A13, A24, A17, A16, A19, Th17;
hence contradiction by A1, A21, A24, A25, Th20; :: thesis: verum