reserve x, y, y1, y2 for object;
reserve V for Z_Module;
reserve W, W1, W2 for Submodule of V;
reserve u, v for VECTOR of V;
reserve i, j, k, n for Element of NAT;

theorem
  for V being finite-rank free Z_Module, W1, W2 being Submodule of V holds
  rank(W1 /\ W2) >= rank W1 + rank W2 - rank V
  proof
    let V be finite-rank free Z_Module, W1, W2 be Submodule of V;
    A1: rank(W1 + W2) <= rank V &
    rank V + (rank(W1 /\ W2) - rank V) = rank(W1 /\ W2) by ZMODUL05:2;
    rank W1 + rank W2 - rank V = rank(W1 + W2) + rank(W1 /\ W2) - rank V
    by ZMODUL06:62
    .= rank(W1 + W2) + (rank(W1 /\ W2) - rank V);
    hence thesis by A1,XREAL_1:6;
  end;
