reserve x,y for object,
  f for Function;

theorem Th43:
  for X,Y being non empty TopSpace for S being Scott
TopAugmentation of InclPoset the topology of Y for W being open Subset of [:X,Y
  :] holds (W, the carrier of X)*graph is continuous Function of X, S
proof
  let X,Y be non empty TopSpace;
  let S be Scott TopAugmentation of InclPoset the topology of Y;
  let W be open Subset of [:X,Y:];
  set f = (W, the carrier of X)*graph;
  reconsider W as Relation of the carrier of X, the carrier of Y by
BORSUK_1:def 2;
A1: dom f = the carrier of X by Def5;
A2: the carrier of InclPoset the topology of Y = the topology of Y & the
RelStr of S = the RelStr of InclPoset the topology of Y by YELLOW_1:1
,YELLOW_9:def 4;
  rng f c= the carrier of S
  proof
    let y be object;
    assume y in rng f;
    then consider x being object such that
A3: x in dom f and
A4: y = f.x by FUNCT_1:def 3;
    reconsider x as Element of X by A3,Def5;
    y = Im(W,x) by A4,Def5;
    then y is open Subset of Y by Th42;
    hence thesis by A2,PRE_TOPC:def 2;
  end;
  then reconsider f as Function of X,S by A1,FUNCT_2:2;
  dom W c= the carrier of X;
  then *graph f = W by Th41;
  hence thesis by Th40;
end;
