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 Th06:
  A <> C & A,B,C are_collinear implies
  (affine-ratio(A,B,C) = 0 iff A = B)
  proof
    assume that
A1: A <> C and
A2: A,B,C are_collinear;
    hereby
      assume affine-ratio(A,B,C) = 0;
      then (B - A) = 0 * (C - A) by A1,A2,Def02
                  .= 0.V by RLVECT_1:10;
      hence A = B by RLVECT_1:21;
    end;
    assume A = B;
    then B - A = 0.V by RLVECT_1:5
              .= 0 * (C - A) by RLVECT_1:10;
    hence affine-ratio(A,B,C) = 0 by A1,A2,Def02;
  end;
