
theorem Th49:
  for W being with_non-empty_element set for a,b being Object of W-CL_category
  for f being set holds f in <^a,b^> iff
  f is infs-preserving 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-CL_category, f be set;
  the carrier of W-CL_category c= the carrier of W-INF(SC)_category by
ALTCAT_2:def 11;
  then reconsider a1 = a, b1 = b as Object of W-INF(SC)_category;
  <^a,b^> = <^a1,b1^> by ALTCAT_2:28;
  hence thesis by Th44;
end;
