 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 LmRank1:
  for V being torsion-free Z_Module,
  v, u being Vector of V st v <> 0.V & u <> 0.V & Lin{v} /\ Lin{u} <> (0).V
  holds rank(Lin{v} + Lin{u}) = 1
  proof
    let V be torsion-free Z_Module,
    v, u be Vector of V such that
    A1: v <> 0.V & u <> 0.V & Lin{v} /\ Lin{u} <> (0).V;
    consider w be Vector of V such that
    A2: w <> 0.V & Lin{v} + Lin{u} = Lin{w} by A1,LmSumMod2;
    w in Lin{w} by ZMODUL02:65,ZFMISC_1:31;
    then reconsider ww = w as Vector of Lin{w};
    A3: ww <> 0.Lin{w} by A2,ZMODUL01:26;
    (Omega).Lin{w} = Lin{ww} by ZMODUL03:20;
    hence thesis by A2,A3,ZMODUL05:5;
  end;
