 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 finite-rank free Z_Module,
  W1, W2 being finite-rank free Subspace of V
  st V is_the_direct_sum_of W1,W2 holds rank V = rank W1 + rank W2
  proof
    let V be finite-rank free Z_Module,
    W1, W2 be finite-rank free Subspace of V such that
    A1: V is_the_direct_sum_of W1,W2;
    rank((Omega).V) = rank(W1) + rank(W2) by ThRankDirectSum,A1;
    hence thesis by ZMODUL05:4;
  end;
