
theorem Th51:
  for W being with_non-empty_element set
  for a, b being Object of W-CL-opp_category for f being set
  holds f in <^a,b^> iff
  ex g being sups-preserving Function of latt a, latt b st g = f &
  UpperAdj g is directed-sups-preserving
proof
  let W be with_non-empty_element set;
  let a,b be Object of W-CL-opp_category, f be set;
  the carrier of W-CL-opp_category c= the carrier of W-SUP(SO)_category
  by ALTCAT_2:def 11;
  then reconsider a1 = a, b1 = b as Object of W-SUP(SO)_category;
  <^a,b^> = <^a1,b1^> by ALTCAT_2:28;
  hence thesis by Th46;
end;
