 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 LmISRank21:
  for V being torsion-free Z_Module,
  W1, W2 being finite-rank free Subspace of V,
  v being Vector of V
  st W1 /\ Lin{v} <> (0).V & W2 /\ Lin{v} <> (0).V
  holds (W1 /\ W2) /\ Lin{v} <> (0).V
  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: W1 /\ Lin{v} <> (0).V & W2 /\ Lin{v} <> (0).V;
    consider u1 be Vector of V such that
    A2: u1 in W1 /\ Lin{v} & u1 <> 0.V by A1,ZMODUL04:24;
    consider u2 be Vector of V such that
    A3: u2 in W2 /\ Lin{v} & u2 <> 0.V by A1,ZMODUL04:24;
    A6: u1 in Lin{v} by A2,ZMODUL01:94;
    then consider iu1 be Element of INT.Ring such that
    A4: u1 = iu1*v by ThLin1;
    u2 in Lin{v} by A3,ZMODUL01:94;
    then consider iu2 be Element of INT.Ring such that
    A5: u2 = iu2*v by ThLin1;
    A7: iu2 <> 0.INT.Ring by A3,A5,ZMODUL01:1;
    A8: iu2 * u1 = (iu2*iu1) * v by VECTSP_1:def 16,A4
    .= iu1 * u2 by A5,VECTSP_1:def 16;
    u1 in W1 by A2,ZMODUL01:94;
    then A9: iu2 * u1 in W1 by ZMODUL01:37;
    u2 in W2 by A3,ZMODUL01:94;
    then iu2 * u1 in W2 by A8,ZMODUL01:37;
    then A10: iu2 * u1 in W1 /\ W2 by A9,ZMODUL01:94;
    iu2 * u1 in Lin{v} by A6,ZMODUL01:37;
    then iu2 * u1 in (W1 /\ W2) /\ Lin{v} by A10,ZMODUL01:94;
    hence thesis by ZMODUL02:66,A2,A7,ZMODUL01:def 7;
  end;
