 reserve X,a,b,c,x,y,z,t for set;
 reserve R for Relation;

theorem UApAdditive:
  for R being non empty RelStr,
      A, B being Subset of R holds
    (UAp R).(A \/ B) = (UAp R).A \/ (UAp R).B
  proof
    let R be non empty RelStr;
    let X, Y be Subset of R;
    set H = UAp R;
    H.(X \/ Y) = UAp (X \/ Y) by ROUGHS_2:def 11
              .= UAp X \/ UAp Y by ROUGHS_2:13
              .= H.X \/ UAp Y by ROUGHS_2:def 11
              .= H.X \/ H.Y by ROUGHS_2:def 11;
    hence thesis;
  end;
