
theorem Th23:
  for X,Y being non empty TopSpace for S being Scott
  TopAugmentation of InclPoset the topology of Y for f1, f2 being Element of
  ContMaps(X, S) st f1 <= f2 holds *graph f1 c= *graph f2
proof
  let X,Y be non empty TopSpace;
  let S be Scott TopAugmentation of InclPoset the topology of Y;
  let f1, f2 be Element of ContMaps(X, S);
  assume
A1: f1 <= f2;
  reconsider F1=f1,F2=f2 as Function of X,S by WAYBEL24:21;
  let a,b be object;
A2: the RelStr of S = the RelStr of InclPoset the topology of Y by
YELLOW_9:def 4;
  f2 is Function of X,S by WAYBEL24:21;
  then
A3: dom f2 = the carrier of X by FUNCT_2:def 1;
  assume
A4: [a,b] in *graph f1;
  then
A5: a in dom f1 & b in f1.a by WAYBEL26:38;
  f1 is Function of X,S by WAYBEL24:21;
  then
A6: dom f1 = the carrier of X by FUNCT_2:def 1;
  then reconsider a9=a as Element of X by A4,WAYBEL26:38;
  F1.a9 is Element of S;
  then f1.a in the carrier of InclPoset the topology of Y by A2;
  then
A7: f1.a in the topology of Y by YELLOW_1:1;
  F2.a9 is Element of S;
  then f2.a in the carrier of InclPoset the topology of Y by A2;
  then
A8: f2.a in the topology of Y by YELLOW_1:1;
  [f1.a9,f2.a9] in the InternalRel of S by A1,WAYBEL24:20;
  then [f1.a,f2.a] in RelIncl the topology of Y by A2,YELLOW_1:1;
  then f1.a c= f2.a by A7,A8,WELLORD2:def 1;
  hence thesis by A6,A3,A5,WAYBEL26:38;
end;
