 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 LmISRank41:
  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(W1) 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 be Basis of W1 such that
    A1: rank(W1 /\ W2) = rank(W1);
    assume ex v being Vector of V st v in I & (W1 /\ W2) /\ Lin{v} = (0).V;
    then consider v be Vector of V such that
    A2: v in I & (W1 /\ W2) /\ Lin{v} = (0).V;
    reconsider II = I as linearly-independent Subset of V
    by ZMODUL03:15,VECTSP_7:def 3;
    A4X: (Omega).W1 = Lin(I) by VECTSP_7:def 3
    .= Lin(II) by ZMODUL03:20;
    then A4: (Omega).W1 = Lin(II \ {v}) + Lin{v} &
    Lin(II \ {v}) /\ Lin{v} = (0).V &
    Lin(II \ {v}) is free & Lin{v} is free & v <> 0.V by A2,ThLin8;
    reconsider LIv = Lin(II \ {v}) as finite-rank free Subspace of V;
    reconsider W1s = (Omega).W1, W2s = (Omega).W2
    as strict finite-rank free Subspace of V by ZMODUL01:42;
    rank(W1s) = rank((W1s /\ W2s) + W1s) by ZMODUL01:112
    .= rank((W1 /\ W2) + W1s) by ZMODUL04:23;
    then A6: rank(W1) = rank((W1 /\ W2) + W1s) by ZMODUL05:4;
    (W1 /\ W2) + W1s = (W1 /\ W2) + (LIv + Lin{v}) by A4X,A2,ThLin8
    .= ((W1 /\ W2) + Lin{v}) + LIv by ZMODUL01:96;
    then (W1 /\ W2) + Lin{v} is Subspace of (W1 /\ W2) + W1s by ZMODUL01:97;
    then A8: rank((W1 /\ W2) + Lin{v}) <= rank(W1) by A6,ZMODUL05:2;
    rank((W1 /\ W2) + Lin{v}) = rank(W1 /\ W2) + rank(Lin{v})
    by A2,ThRankDirectSum
    .= rank(W1) + 1 by A1,A4,LmRank0a;
    hence contradiction by A8,NAT_1:13;
  end;
