 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 ThISRank4:
  for V being torsion-free Z_Module,
  W1, W2 being finite-rank free Subspace of V
  st rank(W1 /\ W2) = rank(W1) holds
  rank(W1 + W2) = rank(W2)
  proof
    let V be torsion-free Z_Module,
    W1, W2 be finite-rank free Subspace of V such that
    A1: rank(W1 /\ W2) = rank(W1);
    set I = the Basis of W1;
    for v being Vector of V st v in I holds (W1 /\ W2) /\ Lin{v} <> (0).V
    by A1,LmISRank41;
    hence thesis by LmISRank42;
  end;
