 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 v <> 0.V & W1 /\ Lin{v} = (0).V & (W1 + W2) /\ Lin{v} = (0).V holds
  rank((W1 + Lin{v}) /\ W2) = rank(W1 /\ W2)
  proof
    let V be torsion-free Z_Module,
    W1, W2 be finite-rank free Subspace of V,
    v be Vector of V such that
    A1: v <> 0.V & W1 /\ Lin{v} = (0).V & (W1 + W2) /\ Lin{v} = (0).V;
    for u being Vector of V st u in (W1 /\ W2) holds
    u in (W1 + Lin{v}) /\ W2
    proof
      let u be Vector of V such that
      B1: u in (W1 /\ W2);
      u in W1 & u in W2 by B1,ZMODUL01:94;
      then u in W1 + Lin{v} & u in W2 by ZMODUL01:93;
      hence thesis by ZMODUL01:94;
    end;
    then (W1 /\ W2) is Subspace of (W1 + Lin{v}) /\ W2 by ZMODUL01:44;
    then A2: rank((W1 + Lin{v}) /\ W2) >= rank(W1 /\ W2) by ZMODUL05:2;
    assume AS: rank((W1 + Lin{v}) /\ W2) <> rank(W1 /\ W2);
    ex u being Vector of V st u in (W1 + Lin{v}) /\ W2 &
    not u in W1 /\ W2
    proof
      assume for u being Vector of V st u in (W1 + Lin{v}) /\ W2
      holds u in W1 /\ W2;
      then (W1 + Lin{v}) /\ W2 is Subspace of W1 /\ W2 by ZMODUL01:44;
      then rank((W1 + Lin{v}) /\ W2) <= rank(W1 /\ W2) by ZMODUL05:2;
      hence contradiction by AS,A2,XXREAL_0:1;
    end;
    then consider u be Vector of V such that
    A4: u in (W1 + Lin{v}) /\ W2 & not u in W1 /\ W2;
    u in W1 + Lin{v} by A4,ZMODUL01:94;
    then consider u1, u2 be Vector of V such that
    A5: u1 in W1 & u2 in Lin{v} & u = u1 + u2 by ZMODUL01:92;
    A6: u in W2 by A4,ZMODUL01:94;
    A7: u2 <> 0.V by A4,A5,A6,ZMODUL01:94;
    A8: -u1 in W1 by A5,ZMODUL01:38;
    u - u1 = u2 + (u1 - u1) by RLVECT_1:28,A5
    .= u2 + 0.V by RLVECT_1:15
    .= u2;
    then u2 in W1 + W2 by A6,A8,ZMODUL01:92;
    hence contradiction by A1,A7,ZMODUL02:66,A5,ZMODUL01:94;
  end;
