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

now

assume ( a <> b & a <> c ) ; :: thesis: a,b // c,d
then ( a <> b & a <> c & a,b // a,c & a,b // a,d ) by A1, Def2;
then ( a <> c & a,b // a,c & a,c // a,d ) by Def1;
then ( a <> c & a,b // a,c & a,c // c,d ) by Th18;
hence a,b // c,d by Th20; :: thesis: verum

end;

hence a,b // c,d by A1, Def2, Th14; :: thesis: verum