 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 LmRank421:
  for V being torsion-free Z_Module, W being finite-rank free Subspace of V,
  u, v being Vector of V st W /\ Lin{v} = (0).V &
  (W + Lin{u}) /\ Lin{v} <> (0).V
  holds W /\ Lin{u} = (0).V
  proof
    let V be torsion-free Z_Module, W be finite-rank free Subspace of V,
    u, v be Vector of V such that
    A1: W /\ Lin{v} = (0).V & (W + Lin{u}) /\ Lin{v} <> (0).V;
    consider x be Vector of V such that
    A2: x in (W + Lin{u}) /\ Lin{v} & x <> 0.V by A1,ZMODUL04:24;
    x in W + Lin{u} by A2,ZMODUL01:94;
    then consider x1, x2 be Vector of V such that
    A3: x1 in W & x2 in Lin{u} & x = x1 + x2 by ZMODUL01:92;
    consider i be Element of INT.Ring such that
    A5: x2 = i*u by A3,ThLin1;
    assume W /\ Lin{u} <> (0).V;
    then consider y be Vector of V such that
    A7: y in W /\ Lin{u} & y <> 0.V by ZMODUL04:24;
    y in Lin{u} by A7,ZMODUL01:94;
    then consider j be Element of INT.Ring such that
    A8: y = j*u by ThLin1;
    A9: j*x1 in W by A3,ZMODUL01:37;
    A10: j*x2 = (j*i)*u by VECTSP_1:def 16,A5
    .= i*y by A8,VECTSP_1:def 16;
    y in W by A7,ZMODUL01:94;
    then j*x2 in W by A10,ZMODUL01:37;
    then j*x1 + j*x2 in W by A9,ZMODUL01:36;
    then A11: j*x in W by A3,VECTSP_1:def 14;
    x in Lin{v} by A2,ZMODUL01:94;
    then j*x in Lin{v} by ZMODUL01:37;
    then A12: j*x in W /\ Lin{v} by A11,ZMODUL01:94;
    j <> 0.INT.Ring by A7,A8,ZMODUL01:1;
    hence contradiction by A1,A12,ZMODUL02:66,A2,ZMODUL01:def 7;
  end;
