reserve x,y for object,
  f for Function;

theorem
  for X,Y being non empty TopSpace for S being Scott TopAugmentation of
InclPoset the topology of Y ex F being Function of InclPoset the topology of [:
  X, Y:], oContMaps(X, S) st F is monotone & for W being open Subset of [:X,Y:]
  holds F.W = (W, the carrier of X)*graph
proof
  let X, Y be non empty TopSpace;
  let S be Scott TopAugmentation of InclPoset the topology of Y;
  deffunc F(Element of the topology of [:X,Y:]) = ($1, the carrier of X)*graph;
  consider F being ManySortedSet of the topology of [:X,Y:] such that
A1: for R being Element of the topology of [:X,Y:] holds F.R = F(R) from
  PBOOLE:sch 5;
A2: rng F c= the carrier of oContMaps(X, 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 the topology of [:X,Y:] by A3;
A5: x is open Subset of [:X,Y:] by PRE_TOPC:def 2;
    y = (x, the carrier of X)*graph by A1,A4;
    then y is continuous Function of X,S by A5,Th43;
    then y is Element of oContMaps(X, S) by Th2;
    hence thesis;
  end;
A6: dom F = the topology of [:X,Y:] by PARTFUN1:def 2;
A7: the carrier of InclPoset the topology of [:X,Y:] = the topology of [:X,Y
  :] by YELLOW_1:1;
  then reconsider
  F as Function of InclPoset the topology of [:X, Y:], oContMaps(X,
  S) by A6,A2,FUNCT_2:2;
  take F;
  thus F is monotone
  proof
    let x,y be Element of InclPoset the topology of [:X,Y:];
    x in the topology of [:X,Y:] & y in the topology of [:X,Y:] by A7;
    then reconsider W1 = x, W2 = y as open Subset of [:X,Y:] by PRE_TOPC:def 2;
    assume x <= y;
    then
A8: W1 c= W2 by YELLOW_1:3;
    F.x = (W1, the carrier of X)*graph & F.y = (W2, the carrier of X)
    *graph by A1,A7;
    hence thesis by A8,Th44;
  end;
  let W be open Subset of [:X,Y:];
  W in the topology of [:X,Y:] by PRE_TOPC:def 2;
  hence thesis by A1;
end;
