reserve a for set;

theorem Th7:
  for L being lower-bounded sup-Semilattice
  for a,b being auxiliary(ii) Relation of L holds
  a /\ b is auxiliary(ii) Relation of L
proof
  let L be with_suprema lower-bounded Poset;
  let a,b be auxiliary(ii) Relation of L;
  reconsider ab = a /\ b as Relation of L;
  for x, y, z, u be Element of L holds
  u <= x & [x,y] in ab & y <= z implies [u,z] in ab
  proof
    let x, y, z, u be Element of L;
    assume that
A1: u <= x and
A2: [x,y] in ab and
A3: y <= z;
A4: [x,y] in a by A2,XBOOLE_0:def 4;
A5: [x,y] in b by A2,XBOOLE_0:def 4;
A6: [u,z] in a by A1,A3,A4,Def4;
    [u,z] in b by A1,A3,A5,Def4;
    hence thesis by A6,XBOOLE_0:def 4;
  end;
  hence thesis by Def4;
end;
