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 that
A1: a <> b and
A2: a,b,c is_collinear and
A3: a,b // c,d ; :: thesis: a,c // b,d

now

A4: a,b // a,c by A2, Def2;
then a,b // c,b by Th18;
then c,b // c,d by A1, A3, Def1;
then A5: b,c // b,d by Th18;
assume A6: b <> c ; :: thesis: a,c // b,d
b,c // a,c by A4, Th18;
hence a,c // b,d by A6, A5, Def1; :: thesis: verum

end;

hence a,c // b,d by A3; :: thesis: verum