reserve x, y, y1, y2 for object;
reserve V for Z_Module;
reserve W, W1, W2 for Submodule of V;
reserve u, v for VECTOR of V;
reserve i, j, k, n for Element of NAT;
reserve V,W for finite-rank free Z_Module;
reserve T for linear-transformation of V,W;

theorem LmStrict11:
  for R being Ring
  for V being LeftMod of R, W being Subspace of V,
      Ws being strict Subspace of V,
  A being object st Ws = (Omega).W holds
  A is Coset of W iff A is Coset of Ws
  proof
    let R be Ring;
    let V be LeftMod of R, W be Subspace of V, Ws be strict Subspace of V,
    A be object such that
    A1: Ws = (Omega).W;
    hereby
      assume A is Coset of W;
      then consider v be Vector of V such that
      B1: A = v + W by VECTSP_4:def 6;
      A = v + Ws by A1,B1,LmStrict11a;
      hence A is Coset of Ws by VECTSP_4:def 6;
    end;
    assume A is Coset of Ws;
    then consider v be Vector of V such that
    B1: A = v + Ws by VECTSP_4:def 6;
    A = v + W by A1,B1,LmStrict11a;
    hence A is Coset of W by VECTSP_4:def 6;
  end;
