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 that
A1: a,b,c is_collinear and
A2: a,b,d is_collinear ; :: thesis: a,b // c,d

now

assume that
A3: a <> b and
A4: a <> c ; :: thesis: a,b // c,d
A5: a,b // a,c by A1, Def2;
a,b // a,d by A2, Def2;
then a,c // a,d by A3, A5, Def1;
then a,c // c,d by Th18;
hence a,b // c,d by A4, A5, Th20; :: thesis: verum

end;

hence a,b // c,d by A2, Def2, Th14; :: thesis: verum