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 Th07:
  A <> C & A,B,C are_collinear implies
  (affine-ratio(A,B,C) = 1 iff B = C)
  proof
    assume that
A1: A <> C and
A2: A,B,C are_collinear;
    hereby
      assume affine-ratio(A,B,C) = 1;
      then (B - A) = 1 * (C - A) by A1,A2,Def02
                  .= C - A by RLVECT_1:def 8;
      hence B = C by RLVECT_1:8;
    end;
    assume B = C;
    then B - A = 1 * (C - A) by RLVECT_1:def 8;
    hence affine-ratio(A,B,C) = 1 by A1,A2,Def02;
  end;
