reserve a for set;

theorem Th10:
  for L being lower-bounded sup-Semilattice
  for a,b being auxiliary Relation of L holds
  a /\ b is auxiliary Relation of L
proof
  let L be with_suprema lower-bounded Poset;
  let a,b be auxiliary Relation of L;
  reconsider ab = a /\ b as Relation of L;
  ab is auxiliary(i) auxiliary(ii) auxiliary(iii) auxiliary(iv)
  by Th6,Th7,Th8,Th9;
  hence thesis;
end;
