reserve x, y for set;

theorem Th12:
  for W being with_non-empty_element set holds the carrier of W
  -UPS_category c= POSETS W
proof
  let W be with_non-empty_element set;
  let x be object;
  assume x in the carrier of W-UPS_category;
  then reconsider x as Object of W-UPS_category;
A1: ex w being non empty set st w in W by SETFAM_1:def 10;
  then
A2: the carrier of latt x in W by Def10;
  latt x = x;
  then x is strict Poset by A1,Def10;
  hence thesis by A2,Def2;
end;
