
theorem  :: Example 2
  for R being non empty void RelStr,
      X being Subset of R holds
    UAp 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) meets X } =
    { x where x is Element of R : {} meets X}
    proof
      thus { x where x is Element of R : Class ({},x) meets X } c=
      { x where x is Element of R : {} meets X}
      proof
        let y be object;
        assume y in { x where x is Element of R : Class ({},x) meets X }; then
        consider z being Element of R such that
A2:     z = y & Class ({},z) meets X;
        thus thesis by A2;
      end;
      let y be object;
      assume y in { x where x is Element of R : {} meets X}; then
      consider z being Element of R such that
A3:   z = y & {} meets X;
      thus thesis by A3;
    end;
    hence thesis by Th3,A1;
  end;
