reserve x, y for set;

theorem
  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-ALG_category iff L is strict complete
  algebraic
proof
  let W be with_non-empty_element set, L be LATTICE such that
A1: the carrier of L in W;
  hereby
    assume L is Object of W-ALG_category;
    then reconsider a = L as Object of W-ALG_category;
    L = latt a & a is Object of W-CONT_category by ALTCAT_2:29;
    hence L is strict complete algebraic by A1,Def12,Th16;
  end;
  assume
A2: L is strict complete algebraic;
  then reconsider a = L as Object of W-CONT_category by A1,Th16;
  latt a = L;
  hence thesis by A2,Def12;
end;
