
theorem LMQ05:
  for X be RealLinearSpace, Y be Subspace of X
  holds CosetSet(X,Y) = CosetSet(RLSp2RVSp X,RLSp2RVSp Y)
  proof
    let X be RealLinearSpace, Y be Subspace of X;
    for x be object holds
    x in CosetSet(X,Y) iff x in CosetSet(RLSp2RVSp X,RLSp2RVSp Y)
    proof
      let x be object;
      hereby
        assume x in CosetSet(X,Y); then
        consider A be Coset of Y such that
        A1: x = A;
        x is Coset of RLSp2RVSp Y by A1,LMQ04;
        hence x in CosetSet(RLSp2RVSp X, RLSp2RVSp Y);
      end;
      assume x in CosetSet(RLSp2RVSp X, RLSp2RVSp Y); then
      consider B be Coset of RLSp2RVSp Y such that
      A2: x = B;
      x is Coset of Y by A2,LMQ04;
      hence x in CosetSet(X,Y);
    end;
    hence thesis by TARSKI:2;
  end;
