
theorem Th35:
  for L be sup-Semilattice for x be Element of L holds downarrow x
  is join-closed
proof
  let L be sup-Semilattice;
  let x be Element of L;
  reconsider x1 = x as Element of L;
  now
    let y,z be Element of L;
    assume that
A1: y in the carrier of subrelstr downarrow x and
A2: z in the carrier of subrelstr downarrow x and
    ex_sup_of {y,z},L;
    z in downarrow x by A2,YELLOW_0:def 15;
    then
A3: z <= x1 by WAYBEL_0:17;
    y in downarrow x by A1,YELLOW_0:def 15;
    then y <= x1 by WAYBEL_0:17;
    then y"\/"z <= x1 by A3,YELLOW_5:9;
    then y"\/"z in downarrow x by WAYBEL_0:17;
    then sup {y,z} in downarrow x by YELLOW_0:41;
    hence sup {y,z} in the carrier of subrelstr downarrow x by YELLOW_0:def 15;
  end;
  then subrelstr downarrow x is join-inheriting;
  hence thesis;
end;
