reserve a,b,r for non unit non zero Real;
reserve X for non empty set,
        x for Tuple of 4,X;
reserve V             for RealLinearSpace,
        A,B,C,P,Q,R,S for Element of V;

theorem
  A <> C & A,B,C are_collinear implies
  A - B = affine-ratio(A,B,C) * (A - C)
  proof
    assume that
A1: A <> C and
A2: A,B,C are_collinear;
    A - B = - (B - A) by RLVECT_1:33
         .= - affine-ratio(A,B,C) * (C - A) by Def02,A1,A2
         .= (- 1) * (affine-ratio(A,B,C) * (C - A)) by RLVECT_1:16
         .= ((- 1) * affine-ratio(A,B,C)) * (C - A) by RLVECT_1:def 7
         .= affine-ratio(A,B,C) * ((- 1) * (C - A)) by RLVECT_1:def 7
         .= affine-ratio(A,B,C) * (-(C - A)) by RLVECT_1:16
         .= affine-ratio(A,B,C) * (A - C) by RLVECT_1:33;
    hence thesis;
  end;
