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 LmStrict3:
  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
  lmultCoset(V, W) = lmultCoset(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 = lmultCoset(V, W);
    set f2 = lmultCoset(V, Ws);
    set C = CosetSet(V,W);
    set Cs = CosetSet(V,Ws);
    A14: CosetSet(V,W) = CosetSet(V,Ws) by AS,LmStrict1;
    now
      let z be Element of R;
      let A 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;
      reconsider As = A as Element of Cs by AS,LmStrict1;
      A21:As = a + Ws by AS,A17,A18,LmStrict11a;
      thus f1.(z,A) = z*a + W by A17,A18,VECTSP10:def 5
      .= z*a+Ws by LmStrict11a,AS
      .= f2.(z,A) by A21,VECTSP10:def 5;
    end;
    hence thesis by A14,BINOP_1:2;
  end;
