reserve a for set;

theorem Th6:
  for L being lower-bounded sup-Semilattice
  for a,b being auxiliary(i) Relation of L holds
  a /\ b is auxiliary(i) Relation of L
proof
  let L be with_suprema lower-bounded Poset;
  let a,b be auxiliary(i) Relation of L;
  reconsider ab = a /\ b as Relation of L;
  for x, y be Element of L holds [x,y] in ab implies x <= y
  proof
    let x, y be Element of L;
    assume [x,y] in ab;
    then [x,y] in a by XBOOLE_0:def 4;
    hence thesis by Def3;
  end;
  hence thesis by Def3;
end;
