reserve x,y for object,
  f for Function;

theorem Th44:
  for X,Y being non empty TopSpace for S being Scott
TopAugmentation of InclPoset the topology of Y for W1, W2 being open Subset of
  [:X,Y:] st W1 c= W2 for f1, f2 being Element of oContMaps(X, S) st f1 = (W1,
  the carrier of X)*graph & f2 = (W2, the carrier of X)*graph holds f1 <= f2
proof
  let X,Y be non empty TopSpace;
  let S be Scott TopAugmentation of InclPoset the topology of Y;
  let W1,W2 be open Subset of [:X,Y:] such that
A1: W1 c= W2;
  let f1,f2 be Element of oContMaps(X, S) such that
A2: f1 = (W1, the carrier of X)*graph & f2 = (W2, the carrier of X) *graph;
  reconsider g1 = f1, g2 = f2 as continuous Function of X, Omega S by Th1;
  S is TopAugmentation of S by YELLOW_9:44;
  then
A3: the RelStr of S = the RelStr of Omega S by WAYBEL25:16;
A4: the RelStr of S = the RelStr of InclPoset the topology of Y by
YELLOW_9:def 4;
  now
    let j be set;
    assume j in the carrier of X;
    then reconsider x = j as Element of X;
    reconsider g1x = g1.x, g2x = g2.x as Element of InclPoset the topology of
    Y by A3,YELLOW_9:def 4;
    take a = g1.x, b = g2.x;
    thus a = g1.j & b = g2.j;
    g1.x = Im(W1,x) & g2.x = Im(W2,x) by A2,Def5;
    then g1.x c= g2.x by A1,RELAT_1:124;
    then g1x <= g2x by YELLOW_1:3;
    hence a <= b by A4,A3,YELLOW_0:1;
  end;
  then g1 <= g2;
  hence thesis by Th3;
end;
