
theorem Th13:  :: Proposition 2 4H
  for R being non empty RelStr,
      X, Y being Subset of R holds
    UAp (X \/ Y) = UAp X \/ UAp Y
  proof
    let R be non empty RelStr;
    let X, Y be Subset of R;
    thus UAp (X \/ Y) c= UAp X \/ UAp Y
    proof
      let y be object;
      assume y in UAp (X \/ Y); then
      consider z being Element of R such that
A1:   z = y & Class (the InternalRel of R, z) meets (X \/ Y);
      Class (the InternalRel of R, z) meets X or
      Class (the InternalRel of R, z) meets Y by A1,XBOOLE_1:70; then
      z in { x where x is Element of R :
      Class (the InternalRel of R, x) meets X} or
      z in { x where x is Element of R :
      Class (the InternalRel of R, x) meets Y};
      hence thesis by A1,XBOOLE_0:def 3;
    end;
    let y be object;
    assume y in UAp X \/ UAp Y; then
    per cases by XBOOLE_0:def 3;
    suppose y in UAp X; then
      consider z being Element of R such that
A2:   z = y & Class (the InternalRel of R, z) meets X;
      Class (the InternalRel of R, z) meets (X \/ Y) by A2,XBOOLE_1:70;
      hence thesis by A2;
    end;
    suppose y in UAp Y; then
      consider z being Element of R such that
A3:   z = y & Class (the InternalRel of R, z) meets Y;
      Class (the InternalRel of R, z) meets (X \/ Y) by A3,XBOOLE_1:70;
      hence thesis by A3;
    end;
  end;
