
theorem LMQ09:
  for V be RealLinearSpace, W be Subspace of V holds
  for A be VECTOR of VectQuot(V,W), v be VECTOR of V, a be Real st A = v + W
  holds a * A = a * v + W
  proof
    let V be RealLinearSpace, W be Subspace of V;
    set X = RLSp2RVSp V;
    set Y = RLSp2RVSp W;
    let A be VECTOR of VectQuot(V,W), v be VECTOR of V, a be Real;
    assume
    A1: A = v + W;
    reconsider v1 = v as Vector of X;
    reconsider A1 = A as Vector of VectQuot (X,Y);
    reconsider a1 = a as Element of F_Real by XREAL_0:def 1;
    A2: A1 = v1 + Y by A1,LMQ03;
    thus a * A = a1 * A1
              .= (a1 * v1) + Y by A2,VECTSP10:25
              .= (a * v) + W by LMQ03;
  end;
