reserve x, y for set;

theorem Th14:
  for W being with_non-empty_element set for L being LATTICE st
  the carrier of L in W holds L is Object of W-UPS_category iff L is strict
  complete
proof
  let W be with_non-empty_element set;
  let L be LATTICE;
  assume
A1: the carrier of L in W;
  L as_1-sorted = L by Def1;
  then L in POSETS W iff L is strict by A1,Def2;
  hence thesis by Th13;
end;
