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 ThStrict1:
  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
  VectQuot(V, W) = VectQuot(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;
    set Z1 = VectQuot(V, W);
    set Z2 = VectQuot(V, Ws);
    A2: the carrier of Z1 = CosetSet(V, W) by VECTSP10:def 6
    .= CosetSet(V, Ws) by A1,LmStrict1
    .= the carrier of Z2 by VECTSP10:def 6;
    A3: the addF of Z1 = addCoset(V, W) by VECTSP10:def 6
    .= addCoset(V, Ws) by A1,LmStrict2
    .= the addF of Z2 by VECTSP10:def 6;
    A4: 0.Z1 = zeroCoset(V, W) by VECTSP10:def 6
    .= zeroCoset(V, Ws) by A1
    .= 0.Z2 by VECTSP10:def 6;
    the lmult of Z1 = lmultCoset(V, W) by VECTSP10:def 6
    .= lmultCoset(V, Ws) by A1,LmStrict3
    .= the lmult of Z2 by VECTSP10:def 6;
    hence thesis by A2,A3,A4;
  end;
