
theorem Th16:
  for W being with_non-empty_element set
  for a, b being Object of W-SUP_category for f being set
  holds f in <^a,b^> iff f is sups-preserving Function of latt a, latt b
proof
  let W be with_non-empty_element set;
  let a,b be Object of W-SUP_category, f be set;
A1: ex a being non empty set st a 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,YELLOW21:def 7;
    hence f is sups-preserving Function of latt a, latt b by A1,A2,Def5;
  end;
  thus thesis by A1,Def5;
end;
