let PCPP be CollProjectiveSpace; :: thesis: for o, a, d, d9, a9, x being Element of PCPP st not o,a,d is_collinear & o,d,d9 is_collinear & d <> d9 & a9,d9,x is_collinear & o,a,a9 is_collinear & o <> a9 holds
x <> d
let o, a, d, d9, a9, x be Element of PCPP; :: thesis: ( not o,a,d is_collinear & o,d,d9 is_collinear & d <> d9 & a9,d9,x is_collinear & o,a,a9 is_collinear & o <> a9 implies x <> d )
assume that
A1: not o,a,d is_collinear and
A2: o,d,d9 is_collinear and
A3: d <> d9 and
A4: a9,d9,x is_collinear and
A5: o,a,a9 is_collinear and
A6: o <> a9 ; :: thesis: x <> d
assume not x <> d ; :: thesis: contradiction
then A7: d,d9,a9 is_collinear by A4, Th3;
d,d9,o is_collinear by A2, Th3;
then d,o,a9 is_collinear by A3, A7, Th4;
then A8: o,a9,d is_collinear by Th3;
o,a9,a is_collinear by A5, Th3;
hence contradiction by A1, A6, A8, Th4; :: thesis: verum