 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 LmSumMod4:
  for V being torsion-free Z_Module, W being finite-rank free Subspace of V,
  v being Vector of V
  st v <> 0.V & W /\ Lin{v} <> (0).V
  holds
  ex u being Vector of V st u <> 0.V & W /\ Lin{v} = Lin{u}
  proof
    let V be torsion-free Z_Module, W be finite-rank free Subspace of V,
    v be Vector of V such that
    A1: v <> 0.V & W /\ Lin{v} <> (0).V;
    rank(W /\ Lin{v}) = 1 by A1,LmRank41;
    then consider uu be Vector of W /\ Lin{v} such that
    A2: uu <> 0.(W /\ Lin{v}) & (Omega).(W /\ Lin{v}) = Lin{uu} by ZMODUL05:5;
    reconsider u = uu as Vector of V by ZMODUL01:25;
    A3: u <> 0.V by A2,ZMODUL01:26;
    (Omega).(W /\ Lin{v}) = Lin{u} by A2,ZMODUL03:20;
    hence thesis by A3;
  end;
