let X, Y be non empty TopSpace; :: thesis: 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
let S be Scott TopAugmentation of InclPoset the topology of Y; :: thesis: for W being open Subset of [:X,Y:] holds (W, the carrier of X) *graph is continuous Function of X,S
let W be open Subset of [:X,Y:]; :: thesis: (W, the carrier of X) *graph is continuous Function of X,S
set f = (W, the carrier of X) *graph ;
reconsider W = W as Relation of the carrier of X, the carrier of Y by BORSUK_1:def 2;
A1: dom ((W, the carrier of X) *graph) = the carrier of X by Def5;
A2: ( the carrier of (InclPoset the topology of Y) = the topology of Y & RelStr(# the carrier of S, the InternalRel of S #) = RelStr(# the carrier of (InclPoset the topology of Y), the InternalRel of (InclPoset the topology of Y) #) ) by YELLOW_1:1, YELLOW_9:def 4;
rng ((W, the carrier of X) *graph) c= the carrier of S

proof

let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in rng ((W, the carrier of X) *graph) or y in the carrier of S )
assume y in rng ((W, the carrier of X) *graph) ; :: thesis: y in the carrier of S
then consider x being object such that
A3: x in dom ((W, the carrier of X) *graph) and
A4: y = ((W, the carrier of X) *graph) . x by FUNCT_1:def 3;
reconsider x = 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 y in the carrier of S by A2, PRE_TOPC:def 2; :: thesis: verum

end;

then reconsider f = (W, the carrier of X) *graph as Function of X,S by A1, FUNCT_2:2;
dom W c= the carrier of X ;
then *graph f = W by Th41;
hence (W, the carrier of X) *graph is continuous Function of X,S by Th40; :: thesis: verum