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 LmStrict1:
  for R being Ring
  for V being LeftMod of R, W being Subspace of V,
      Ws being strict Subspace of V
  st Ws = (Omega).W holds
  CosetSet(V, W) = CosetSet(V, Ws)
  proof
    let R be Ring;
    let V be LeftMod of R, W be Subspace of V, Ws be strict Subspace of V
    such that
    A1: Ws = (Omega).W;
    for A being object holds A in CosetSet(V, W) iff A in CosetSet(V, Ws)
    proof
      let A be object;
      hereby
        assume A in CosetSet(V, W);
        then consider AA be Coset of W such that
        C1: A = AA;
        AA is Coset of Ws by A1,LmStrict11;
        hence A in CosetSet(V, Ws) by C1;
      end;
      assume A in CosetSet(V, Ws);
      then consider AA be Coset of Ws such that
      C1: A = AA;
      AA is Coset of W by A1,LmStrict11;
      hence A in CosetSet(V, W) by C1;
    end;
    hence thesis by TARSKI:2;
  end;
