
theorem Th41:
  for L be sup-Semilattice for x be Element of L holds wayabove x
  is join-closed
proof
  let L be sup-Semilattice;
  let x be Element of L;
  now
    let y,z be Element of L;
    assume that
A1: y in the carrier of subrelstr wayabove x and
    z in the carrier of subrelstr wayabove x and
    ex_sup_of {y,z},L;
    y in wayabove x by A1,YELLOW_0:def 15;
    then
A2: y >> x by WAYBEL_3:8;
    y"\/"z >= y by YELLOW_0:22;
    then y"\/"z >> x by A2,WAYBEL_3:2;
    then y"\/"z in wayabove x by WAYBEL_3:8;
    then sup {y,z} in wayabove x by YELLOW_0:41;
    hence sup {y,z} in the carrier of subrelstr wayabove x by YELLOW_0:def 15;
  end;
  then subrelstr wayabove x is join-inheriting;
  hence thesis;
end;
