
theorem LMQ03:
  for X be RealLinearSpace,
      Y be Subspace of X,
      v be Element of X,
      v1 be Element of RLSp2RVSp X
  st v = v1
  holds v + Y = v1 + RLSp2RVSp Y
  proof
    let X be RealLinearSpace,
        Y be Subspace of X,
        v be Element of X,
        v1 be Element of RLSp2RVSp X;
    set V = RLSp2RVSp X;
    set W = RLSp2RVSp Y;
    assume
    A1: v = v1;
    for x be object holds x in v+Y iff x in v1 + W
    proof
      let x be object;
      hereby
        assume x in v+Y; then
        consider y be Point of X such that
        A2: x = v + y & y in Y;
        reconsider y1 = y as Point of V;
        A3: y1 in W by A2;
        v + y = v1 + y1 by A1;
        hence x in v1 + W by A2,A3;
      end;
      assume x in v1 + W; then
      consider y1 be Element of V such that
      A4: x = v1 + y1 & y1 in W;
      reconsider y = y1 as Point of X;
      A5: y in Y by A4;
      v + y = v1 + y1 by A1;
      hence x in v + Y by A4,A5;
    end;
    hence thesis by TARSKI:2;
  end;
