
theorem Th16:  :: Proposition 2 8LH
  for R being non empty RelStr,
      X being Subset of R holds
    LAp (X`) = (UAp X)`
  proof
    let R be non empty RelStr;
    let X be Subset of R;
    thus LAp (X`) c= (UAp X)`
    proof
      let y be object;
      assume y in LAp (X`); then
      consider z being Element of R such that
A1:   z = y & Class (the InternalRel of R, z) c= X`;
      not z in { x where x is Element of R :
        Class (the InternalRel of R, x) meets X }
      proof
        assume z in { x where x is Element of R :
          Class (the InternalRel of R, x) meets X }; then
        consider t being Element of R such that
A2:     t = z & Class (the InternalRel of R, t) meets X;
        thus contradiction by A1,SUBSET_1:23,A2;
      end;
      hence thesis by A1,XBOOLE_0:def 5;
    end;
    let y be object;
    assume
A3: y in (UAp X)`;
    y in [#]R & not y in UAp X by XBOOLE_0:def 5,A3; then
    not (Class (the InternalRel of R, y) meets X); then
    Class (the InternalRel of R, y) c= X` by SUBSET_1:23;
    hence thesis by A3;
  end;
