reserve x, y for set;

theorem Th19:
  for W being with_non-empty_element set for a,b being Object of W
  -ALG_category for f being set holds f in <^a,b^> iff f is
  directed-sups-preserving Function of latt a, latt b
proof
  let W be with_non-empty_element set;
  let a,b be Object of W-ALG_category, f be set;
  reconsider a1 = a, b1 = b as Object of W-CONT_category by ALTCAT_2:29;
  <^a,b^> = <^a1,b1^> by ALTCAT_2:28;
  hence thesis by Th18;
end;
