
theorem LMQ11:
  for V be RealLinearSpace, W be Subspace of V holds
  for A1,A2 be VECTOR of VectQuot(V,W),
      v1,v2 be VECTOR of V
  st A1 = v1 + W & A2 = v2 + W
  holds A1 + A2 = v1 + v2 + W
  proof
    let V be RealLinearSpace, W be Subspace of V;
    set X = RLSp2RVSp V;
    set Y = RLSp2RVSp W;
    let A1,A2 be VECTOR of VectQuot(V,W),
        v1,v2 be VECTOR of V;
    assume
    A1: A1 = v1 + W & A2 = v2 + W;
    reconsider x1 = v1, x2 = v2 as Vector of X;
    reconsider B1 = A1, B2 = A2 as Vector of VectQuot(X,Y);
    A2: B1 = x1 + Y & B2 = x2 + Y by A1,LMQ03;
    thus A1 + A2 = B1 + B2
                .= (x1 + x2) + Y by A2,VECTSP10:26
                .= (v1 + v2) + W by LMQ03;
  end;
