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
  for a being non zero Real st P,Q,R are_collinear & P <> R holds
  affine-ratio(P,Q,R) = affine-ratio(a * P,a * Q,a * R)
  proof
    let a be non zero Real;
    assume
A1: P,Q,R are_collinear & P <> R;
    reconsider aP = a * P, aQ = a * Q, aR = a * R as Element of V;
    now
      thus aP <> aR by A1,RLVECT_1:36;
      Q in Line(P,R) by A1,RLTOPSP1:80;
      then Q in the set of all (1 - l) * P + l * R where l is Real
        by RLTOPSP1:def 14;
      then consider l be Real such that
A2:   Q = (1 - l) * P + l * R;
      reconsider aL = Line(aP,aR) as line of V;
      H1: aP in aL & aR in aL by RLTOPSP1:72;
      aQ = a * ((1 - l) * P) + a * (l * R) by A2,RLVECT_1:def 5
          .= (a * (1 - l)) * P + a * (l * R) by RLVECT_1:def 7
          .= (a * (1 - l)) * P + (a * l) * R by RLVECT_1:def 7
          .= (1 - l) * (a * P) + (a * l) * R by RLVECT_1:def 7
          .= (1 - l) * aP + l * aR by RLVECT_1:def 7;
      then aQ in the set of all (1 - l) * aP + l * aR where l is Real;
      then aQ in aL by RLTOPSP1:def 14;
      hence aP,aQ,aR are_collinear by H1;
    end;
    then aQ - aP = affine-ratio(aP,aQ,aR) * (aR - aP) by Def02;
    then a * (Q - P) = affine-ratio(aP,aQ,aR) * (aR - aP) by RLVECT_1:34;
    then a * (Q - P) = affine-ratio(aP,aQ,aR) * (a * (R - P))
      by RLVECT_1:34;
    then a * (Q - P) = (affine-ratio(aP,aQ,aR) * a) * (R - P)
      by RLVECT_1:def 7;
    then a * (Q - P) = a * (affine-ratio(aP,aQ,aR) * (R - P))
      by RLVECT_1:def 7;
    then Q - P = affine-ratio(aP,aQ,aR) * (R - P) by RLVECT_1:36;
    hence thesis by A1,Def02;
  end;
