 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 ThRankS5:
  for V being torsion-free Z_Module,
  W1, W2, W3 being finite-rank free Subspace of V
  st rank(W1 + W2) = rank(W2) & W3 is Subspace of W1
  holds rank(W3 + W2) = rank(W2)
  proof
    let V be torsion-free Z_Module,
    W1, W2, W3 be finite-rank free Subspace of V such that
    A1: rank(W1 + W2) = rank(W2) & W3 is Subspace of W1;
    for v being Vector of V st v in W3 + W2 holds v in W1 + W2
    proof
      let v be Vector of V such that
      B1: v in W3 + W2;
      consider v1, v2 be Vector of V such that
      B2: v1 in W3 & v2 in W2 & v = v1 + v2 by B1,ZMODUL01:92;
      v1 in W1 by A1,B2,ZMODUL01:23;
      hence thesis by B2,ZMODUL01:92;
    end;
    then W3 + W2 is Subspace of W1 + W2 by ZMODUL01:44;
    then A2: rank(W3 + W2) <= rank(W1 + W2) by ZMODUL05:2;
    W2 is Subspace of W3 + W2 by ZMODUL01:97;
    then rank(W2) <= rank(W3 + W2) by ZMODUL05:2;
    hence thesis by A1,A2,XXREAL_0:1;
  end;
