let FCPS be up-3-dimensional CollProjectiveSpace; :: thesis: for o, a, d, d', a', s being Element of st not o,a,d is_collinear & o,d,d' is_collinear & d <> d' & a',d',s is_collinear & o,a,a' is_collinear & o <> a' holds
s <> d
let o, a, d, d', a', s be Element of ; :: thesis: ( not o,a,d is_collinear & o,d,d' is_collinear & d <> d' & a',d',s is_collinear & o,a,a' is_collinear & o <> a' implies s <> d )
assume that
A1: not o,a,d is_collinear and
A2: o,d,d' is_collinear and
A3: d <> d' and
A4: a',d',s is_collinear and
A5: o,a,a' is_collinear and
A6: o <> a' ; :: thesis: s <> d
assume not s <> d ; :: thesis: contradiction
then A7: d,d',a' is_collinear by A4, Th1;
d,d',o is_collinear by A2, Th1;
then d,o,a' is_collinear by A3, A7, COLLSP:11;
then A8: o,a',d is_collinear by Th1;
o,a',a is_collinear by A5, Th1;
hence contradiction by A1, A6, A8, COLLSP:11; :: thesis: verum