 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 torsion-free Z_Module,
  W1, W2 being finite-rank free Subspace of V, I being Basis of W1
  st rank(W1 + W2) = rank(W2)
  holds (for v being Vector of V st v in I holds (W1 /\ W2) /\ Lin{v} <> (0).V)
  proof
    let V be torsion-free Z_Module,
    W1, W2 be finite-rank free Subspace of V, I being Basis of W1 such that
    A1: rank(W1 + W2) = rank(W2);
    thus
    for v being Vector of V st v in I holds (W1 /\ W2) /\ Lin{v} <> (0).V
    proof
      let v be Vector of V such that
      C1: v in I;
      reconsider II = I as linearly-independent Subset of V
      by ZMODUL03:15,VECTSP_7:def 3;
      v in II by C1;
      then C2: v <> 0.V by ZMODUL02:57;
      C3: W1 /\ Lin{v} <> (0).V
      proof
        D1: v in W1 by C1;
        D2: v in Lin{v} by ZMODUL02:65,ZFMISC_1:31;
        v in II by C1;
        then v <> 0.V by ZMODUL02:57;
        hence thesis by D2,ZMODUL02:66,D1,ZMODUL01:94;
      end;
      W2 /\ Lin{v} <> (0).V
      proof
        Lin{v} is Subspace of Lin(II) by ZMODUL02:70,C1,ZFMISC_1:31;
        then Lin{v} is Subspace of Lin(I) by ZMODUL03:20;
        then Lin{v} is Subspace of W1 by ZMODUL01:42;
        then D1: rank(Lin{v} + W2) = rank(W2) by A1,ThRankS5;
        assume W2 /\ Lin{v} = (0).V;
        then rank(Lin{v} + W2) = rank(Lin{v}) + rank(W2) by ThRankDirectSum
        .= rank(W2) + 1 by C2,LmRank0a;
        hence contradiction by D1;
      end;
      hence thesis by C3,LmISRank21;
    end;
  end;
