reserve a for set;

theorem Th9:
  for L being lower-bounded sup-Semilattice
  for a,b being auxiliary(iv) Relation of L holds
  a /\ b is auxiliary(iv) Relation of L
proof
  let L be with_suprema lower-bounded Poset;
  let a,b be auxiliary(iv) Relation of L;
  reconsider ab = a /\ b as Relation of L;
  for x be Element of L holds [Bottom L,x] in ab
  proof
    let x be Element of L;
A1: [Bottom L,x] in a by Def6;
    [Bottom L,x] in b by Def6;
    hence thesis by A1,XBOOLE_0:def 4;
  end;
  hence thesis by Def6;
end;
