 reserve V for Z_Module;
 reserve W for Subspace of V;
 reserve v, u for Vector of V;
 reserve i for Element of INT.Ring;

theorem
  for V being Z_Module, W1, W2, W3 being finite-rank free Subspace of V,
  a being Element of INT.Ring st a <> 0.INT.Ring & W3 = a (*) W1 holds
  rank(W3 /\ W2) = rank(W1 /\ W2)
  proof
    let V be Z_Module, W1, W2, W3 be finite-rank free Subspace of V,
    a be Element of INT.Ring such that
    A1: a <> 0.INT.Ring & W3 = a (*) W1;
    W3 /\ W2 is Subspace of W1 /\ W2
    proof
      W3 /\ W2 is Subspace of W3 by ZMODUL01:105;
      then B2: W3 /\ W2 is Subspace of W1 by A1,ZMODUL01:42;
      W3 /\ W2 is Subspace of W2 by ZMODUL01:105;
      hence thesis by B2,ZMODUL02:75;
    end;
    then A2: rank(W3 /\ W2) <= rank(W1 /\ W2) by ZMODUL05:2;
    a (*) (W1 /\ W2) is Subspace of W3 /\ W2
    proof
      reconsider WX = a (*) (W1 /\ W2) as Subspace of V by ZMODUL01:42;
      for v being Vector of V st v in WX holds
      v in W3 /\ W2
      proof
        let v be Vector of V such that
        C1: v in WX;
        consider vx be Vector of (W1 /\ W2) such that
        C2: v = a * vx by C1;
        reconsider vvx = vx as Vector of V by ZMODUL01:25;
        CX: vvx in W1 /\ W2;
        then vvx in W1 by ZMODUL01:94;
        then reconsider vvvx = vvx as Vector of W1;
        a * vvvx in a * W1;
        then a * vvx in W3 by A1,ZMODUL01:29;
        then C3: v in W3 by C2,ZMODUL01:29;
        vvx in W2 by CX,ZMODUL01:94;
        then a * vvx in W2 by ZMODUL01:37;
        then v in W2 by C2,ZMODUL01:29;
        hence thesis by C3,ZMODUL01:94;
      end;
      hence thesis by ZMODUL01:44;
    end;
    then rank(a (*) (W1 /\ W2)) <= rank(W3 /\ W2) by ZMODUL05:2;
    then rank(W1 /\ W2) <= rank(W3 /\ W2) by A1,ThRankS1;
    hence thesis by A2,XXREAL_0:1;
  end;
