
theorem  :: Example 2
  for R being non empty void RelStr,
      X being Subset of R holds
    LAp X = [#]R
  proof
    let R be non empty void RelStr,
        X be Subset of R;
A1: the InternalRel of R = {} by YELLOW_3:def 3;
    { x where x is Element of R : Class ({}, x) c= X } =
    { x where x is Element of R : {} c= X }
    proof
      thus { x where x is Element of R : Class ({}, x) c= X } c=
      { x where x is Element of R : {} c= X }
      proof
        let y be object;
        assume y in { x where x is Element of R : Class ({}, x) c= X }; then
        consider z being Element of R such that
A2:     z = y & Class ({}, z) c= X;
        thus thesis by A2;
      end;
      let y be object;
      assume y in { x where x is Element of R : {} c= X }; then
      consider z being Element of R such that
A3:   z = y & {} c= X;
      Class ({},z) c= X;
      hence thesis by A3;
    end;
    hence thesis by Th2,A1;
  end;
