 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 LmRank41:
  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
  rank(W /\ Lin{v}) = 1
  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;
    A2: rank(Lin{v}) = 1 by A1,LmRank0a;
    A3: W /\ Lin{v} is Subspace of Lin{v} by ZMODUL01:105;
    rank(W /\ Lin{v}) <> 0
    proof
      assume rank(W /\ Lin{v}) = 0;
      then (Omega).(W /\ Lin{v}) = (0).(W /\ Lin{v}) by ZMODUL05:1
      .= (0).V by ZMODUL01:51;
      hence contradiction by A1;
    end;
    hence rank(W /\ Lin{v}) = 1 by A3,NAT_1:25,ZMODUL05:2,A2;
  end;
