 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, v being Vector of V
  st rank(W1 /\ W2) = rank(W1) & (W1 + W2) /\ Lin{v} <> (0).V holds
  W2 /\ Lin{v} <> (0).V
  proof
    let V be torsion-free Z_Module;
    defpred P[Nat] means
    for W1, W2 being finite-rank free Subspace of V, v being Vector of V
    st rank(W1 /\ W2) = rank(W1) & (W1 + W2) /\ Lin{v} <> (0).V
    & rank(W1) = $1 holds W2 /\ Lin{v} <> (0).V;
    A1: P[0]
    proof
      let W1, W2 be finite-rank free Subspace of V, v be Vector of V;
      assume
      B1: rank(W1 /\ W2) = rank(W1) & (W1 + W2) /\ Lin{v} <> (0).V
      & rank(W1) = 0; then
      B2: (Omega).W1 = (0).W1 by ZMODUL05:1
      .= (0).V by ZMODUL01:51;
      reconsider WW2 = (Omega).W2 as strict Subspace of V by ZMODUL01:42;
      B3: W1 + W2 = (0).V + WW2 by B2,ZMODUL04:22
      .= WW2 by ZMODUL01:99;
      (Omega).Lin{v} = Lin{v};
      hence thesis by B3,ZMODUL04:23,B1;
    end;
    A2: for n being Nat st P[n] holds P[n+1]
    proof
      let n be Nat such that
      B1: P[n];
      let W1, W2 be finite-rank free Subspace of V, v be Vector of V;
      assume
      B2: rank(W1 /\ W2) = rank(W1) & (W1 + W2) /\ Lin{v} <> (0).V
      & rank(W1) = n+1; then
      consider I be finite Subset of V such that
      B3: I is finite Subset of W1 &
      I is linearly-independent & Lin(I) = (Omega).W1
      & card(I) = n+1 by LmFree2;
      BX1: I is Basis of W1
      proof
        reconsider II = I as Subset of W1 by B3;
        (Omega).W1 = Lin(II) by ZMODUL03:20,B3;
        hence thesis by VECTSP_7:def 3,B3,ZMODUL03:16;
      end;
      I is non empty by B3;
      then consider u be object such that
      B4: u in I by XBOOLE_0:def 1;
      reconsider u as Vector of V by B4;
      B5: (Omega).W1 = Lin(I \ {u}) + Lin{u} & u <> 0.V by B3,B4,ThLin8;
      set Iu = I \ {u};
      {u} is Subset of I by B4,SUBSET_1:41;
      then B7: card(Iu) = n+1 - card({u}) by B3,CARD_2:44
      .= n+1 - 1 by CARD_1:30
      .= n;
      reconsider Iu as finite Subset of V;
      B8: Iu is linearly-independent by B3,XBOOLE_1:36,ZMODUL02:56;
      reconsider LIu = Lin(Iu) as strict finite-rank free Subspace of V;
      BX2: Iu is Basis of LIu by B8,ThLin7;
      C1: for v being Vector of V st v in I holds W2 /\ Lin{v} <> (0).V
      proof
        let v be Vector of V such that
        D1: v in I;
        (W1 /\ W2) /\ Lin{v} = W1 /\ (W2 /\ Lin{v}) by ZMODUL01:104;
        hence (W2 /\ Lin{v}) <> (0).V by ZMODUL01:107,B2,BX1,D1,LmISRank41;
      end;
      B10: rank(LIu /\ W2) = rank(LIu)
      proof
        C2: for v being Vector of V st v in Iu holds
        W2 /\ Lin{v} <> (0).V by C1,ZFMISC_1:56;
        for v being Vector of V st v in Iu holds
        (LIu /\ W2) /\ Lin{v} <> (0).V
        proof
          let v be Vector of V such that
          D1: v in Iu;
          v in LIu & v in Lin{v} by D1,ZMODUL02:65,ZFMISC_1:31;
          then D2: v in LIu /\ Lin{v} by ZMODUL01:94;
          consider iv be Vector of V such that
          D3: iv in W2 /\ Lin{v} & iv <> 0.V by ZMODUL04:24,C2,D1;
          iv in Lin{v} by D3,ZMODUL01:94;
          then consider i be Element of INT.Ring such that
          D4: iv = i * v by ThLin1;
          D5: iv in LIu /\ Lin{v} by D2,D4,ZMODUL01:37;
          iv in W2 by D3,ZMODUL01:94;
          then W2 /\ (LIu /\ Lin{v}) <> (0).V by D3,ZMODUL02:66,D5,ZMODUL01:94;
          hence thesis by ZMODUL01:104;
        end;
        hence thesis by BX2,ThISRank2;
      end;
      (LIu + W2) /\ Lin{v} <> (0).V
      proof
        assume C1: (LIu + W2) /\ Lin{v} = (0).V;
        reconsider WW1 = (Omega).W1 as strict Subspace of V by ZMODUL01:42;
        reconsider WW2 = (Omega).W2 as strict Subspace of V by ZMODUL01:42;
        C2: (Omega).(LIu + Lin{u}) = LIu + Lin{u};
        C3: (LIu + W2) + Lin{u} = W2 + (LIu + Lin{u}) by ZMODUL01:96
        .= WW2 + (LIu + Lin{u}) by C2,ZMODUL04:22
        .= WW2 + WW1 by B3,B4,ThLin8
        .= W2 + W1 by ZMODUL04:22;
        then (LIu + W2) /\ Lin{u} = (0).V by C1,LmRank421,B2;
        then C4: rank((LIu + W2) + Lin{u}) = rank(LIu + W2) + rank(Lin{u})
        by ThRankDirectSum
        .= rank(W2) + rank(Lin{u}) by B10,ThISRank4
        .= rank(W2) + 1 by B5,LmRank0a;
        rank((LIu + W2) + Lin{u}) = rank(W2) by B2,ThISRank4,C3;
        hence contradiction by C4;
      end;
      hence thesis by B1,B10,B7,ZMODUL03:def 5,BX2;
    end;
    A3: for n being Nat holds P[n] from NAT_1:sch 2(A1,A2);
    let W1, W2 be finite-rank free Subspace of V, v be Vector of V;
    assume rank(W1 /\ W2) = rank(W1) & (W1 + W2) /\ Lin{v} <> (0).V;
    hence thesis by A3;
  end;
