reserve x, y for set;

theorem Th13:
  for W being with_non-empty_element set for x holds x is Object
  of W-UPS_category iff x is complete LATTICE & x in POSETS W
proof
  let W be with_non-empty_element set;
  let x;
  hereby
    assume x is Object of W-UPS_category;
    then reconsider a = x as Object of W-UPS_category;
    latt a = x;
    hence x is complete LATTICE;
    a in the carrier of W-UPS_category & the carrier of W-UPS_category c=
    POSETS W by Th12;
    hence x in POSETS W;
  end;
  assume x is complete LATTICE;
  then reconsider L = x as complete LATTICE;
  assume x in POSETS W;
  then
A1: the carrier of L as_1-sorted in W & L is strict by Def2;
  L as_1-sorted = L by Def1;
  hence thesis by A1,Def10;
end;
