
theorem
  for W being with_non-empty_element set for L being LATTICE holds
  L is Object of W-SUP(SO)_category iff
  L is strict complete & the carrier of L in W
proof
  let W be with_non-empty_element set;
  let L be LATTICE;
  the carrier of W-SUP(SO)_category c= the carrier of W-SUP_category
  by ALTCAT_2:def 11;
  then L in the carrier of W-SUP(SO)_category implies
  L is Object of W-SUP_category;
  then L is Object of W-SUP(SO)_category iff L is Object of W-SUP_category
  by Def11;
  hence thesis by Th15;
end;
