
theorem LMQ04:
  for X be RealLinearSpace, Y be Subspace of X holds
  for x be object holds x is Coset of Y iff x is Coset of RLSp2RVSp Y
  proof
    let X be RealLinearSpace, Y be Subspace of X;
    let x be object;
    hereby
      assume x is Coset of Y; then
      consider v be Element of X such that
      A1: x = v + Y by RLSUB_1:def 6;
      reconsider v1 = v as Element of RLSp2RVSp X;
      x = v1 + RLSp2RVSp Y by A1,LMQ03;
      hence x is Coset of RLSp2RVSp Y by VECTSP_4:def 6;
    end;
    assume x is Coset of RLSp2RVSp Y; then
    consider v1 be Element of RLSp2RVSp X such that
    A2: x = v1 + RLSp2RVSp Y by VECTSP_4:def 6;
    reconsider v = v1 as Element of X;
    x = v + Y by A2,LMQ03;
    hence x is Coset of Y by RLSUB_1:def 6;
  end;
