
theorem Th37:
  for L be sup-Semilattice for x be Element of L holds uparrow 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 uparrow x and
    z in the carrier of subrelstr uparrow x and
    ex_sup_of {y,z},L;
    y in uparrow x by A1,YELLOW_0:def 15;
    then
A2: y >= x1 by WAYBEL_0:18;
    y"\/"z >= y by YELLOW_0:22;
    then y"\/"z >= x1 by A2,YELLOW_0:def 2;
    then y"\/"z in uparrow x by WAYBEL_0:18;
    then sup {y,z} in uparrow x by YELLOW_0:41;
    hence sup {y,z} in the carrier of subrelstr uparrow x by YELLOW_0:def 15;
  end;
  then subrelstr uparrow x is join-inheriting;
  hence thesis;
end;
