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 LmStrict2:
  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
  addCoset(V, W) = addCoset(V, Ws)
  proof
    let R be Ring;
    let V be LeftMod of R, W be Subspace of V, Ws be strict Subspace of V;
    assume AS: Ws = (Omega).W;
    set f1 = addCoset(V, W);
    set f2 = addCoset(V, Ws);
    set C = CosetSet(V,W);
    set Cs = CosetSet(V,Ws);
    A14: CosetSet(V,W) = CosetSet(V,Ws) by AS,LmStrict1;
    now
      let A, B be Element of C;
      A in C;
      then consider A1 be Coset of W such that
      A17: A1 = A;
      consider a be Vector of V such that
      A18: A1 = a+W by VECTSP_4:def 6;
      B in C;
      then consider B1 be Coset of W such that
      A19: B1 = B;
      consider b be Vector of V such that
      A20: B1 = b+W by VECTSP_4:def 6;
      reconsider As = A as Element of Cs by AS,LmStrict1;
      A21: As = a+Ws by AS,A17,A18,LmStrict11a;
      reconsider Bs = B as Element of Cs by AS,LmStrict1;
      A22: Bs = b+Ws by AS,A19,A20,LmStrict11a;
      thus f1.(A,B) = a+b + W by A17,A19,A18,A20,VECTSP10:def 3
      .= a+b+Ws by LmStrict11a,AS
      .= f2.(A,B) by A21,A22,VECTSP10:def 3;
    end;
    hence thesis by A14,BINOP_1:2;
  end;
