reserve a for set;

theorem Th8:
  for L being lower-bounded sup-Semilattice
  for a,b being auxiliary(iii) Relation of L holds
  a /\ b is auxiliary(iii) Relation of L
proof
  let L be with_suprema lower-bounded Poset;
  let a,b be auxiliary(iii) Relation of L;
  reconsider ab = a /\ b as Relation of L;
  for x, y, z be Element of L holds
  [x,z] in ab & [y,z] in ab implies [(x "\/" y),z] in ab
  proof
    let x, y, z be Element of L;
    assume that
A1: [x,z] in ab and
A2: [y,z] in ab;
A3: [x,z] in a by A1,XBOOLE_0:def 4;
A4: [x,z] in b by A1,XBOOLE_0:def 4;
A5: [y,z] in a by A2,XBOOLE_0:def 4;
A6: [y,z] in b by A2,XBOOLE_0:def 4;
A7: [(x "\/" y),z] in a by A3,A5,Def5;
    [(x "\/" y),z] in b by A4,A6,Def5;
    hence thesis by A7,XBOOLE_0:def 4;
  end;
  hence thesis by Def5;
end;
