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

now

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

end;

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