reserve x, y for set;

theorem Th15:
  for W being with_non-empty_element set for a,b being Object of W
  -UPS_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-UPS_category;
  let f be set;
A1: ex w being non empty set st w in W by SETFAM_1:def 10;
  hereby
    assume
A2: f in <^a,b^>;
    then reconsider g = f as Morphism of a,b;
    f = @g by A2,Def7;
    hence f is directed-sups-preserving Function of latt a, latt b by A1,A2
,Def10;
  end;
  thus thesis by A1,Def10;
end;
