
theorem Th39:
  for L be sup-Semilattice for x be Element of L holds waybelow 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 waybelow x and
A2: z in the carrier of subrelstr waybelow x and
    ex_sup_of {y,z},L;
    z in waybelow x by A2,YELLOW_0:def 15;
    then
A3: z << x by WAYBEL_3:7;
    y in waybelow x by A1,YELLOW_0:def 15;
    then y << x by WAYBEL_3:7;
    then y"\/"z << x by A3,WAYBEL_3:3;
    then y"\/"z in waybelow x by WAYBEL_3:7;
    then sup {y,z} in waybelow x by YELLOW_0:41;
    hence sup {y,z} in the carrier of subrelstr waybelow x by YELLOW_0:def 15;
  end;
  then subrelstr waybelow x is join-inheriting;
  hence thesis;
end;
