let SAS be Semi_Affine_Space; :: thesis: for a, b, c, d being Element of SAS st a <> b & a,b,c is_collinear & a,b // c,d holds
c,b // c,d
let a, b, c, d be Element of SAS; :: thesis: ( a <> b & a,b,c is_collinear & a,b // c,d implies c,b // c,d )
assume A1: ( a <> b & a,b,c is_collinear & a,b // c,d ) ; :: thesis: c,b // c,d

now

assume a <> c ; :: thesis: c,b // c,d
then ( a <> c & a <> b & a,b // a,c & a,b // c,d ) by A1, Def2;
then ( a <> c & a,c // c,b & a,c // c,d ) by Def1, Th18;
hence c,b // c,d by Def1; :: thesis: verum

end;

hence c,b // c,d by A1; :: thesis: verum