
theorem Th44:
  for W being with_non-empty_element set
  for a, b being Object of W-INF(SC)_category for f being set
  holds f in <^a,b^> iff
  f is directed-sups-preserving infs-preserving Function of latt a, latt b
proof
  let W be with_non-empty_element set;
  let a,b be Object of W-INF(SC)_category, f be set;
  the carrier of W-INF(SC)_category c= the carrier of W-INF_category by
ALTCAT_2:def 11;
  then reconsider a1 = a, b1 = b as Object of W-INF_category;
  hereby
    assume
A1: f in <^a,b^>;
A2: <^a,b^> c= <^a1,b1^> by ALTCAT_2:31;
    then reconsider g = f as Morphism of a1,b1 by A1;
    f = @g by A1,A2,YELLOW21:def 7;
    hence f is directed-sups-preserving infs-preserving Function
    of latt a, latt b by A1,A2,Def10,Th14;
  end;
  assume f is directed-sups-preserving infs-preserving Function
  of latt a, latt b;
  then reconsider f as
  directed-sups-preserving infs-preserving Function of latt a, latt b;
A3: f in <^a1,b1^> by Th14;
  reconsider g = f as Morphism of a1,b1 by Th14;
  f = @g by A3,YELLOW21:def 7;
  hence thesis by A3,Def10;
end;
