let Y be T_0-TopSpace; :: thesis: ( InclPoset the topology of Y is continuous iff for X being non empty TopSpace
for f being continuous Function of X,(Sigma (InclPoset the topology of Y)) holds *graph f is open Subset of [:X,Y:] )

hereby :: thesis: ( ( for X being non empty TopSpace
for f being continuous Function of X,(Sigma (InclPoset the topology of Y)) holds *graph f is open Subset of [:X,Y:] ) implies InclPoset the topology of Y is continuous )
assume InclPoset the topology of Y is continuous ; :: thesis: for X being non empty TopSpace
for f being continuous Function of X,(Sigma (InclPoset the topology of Y)) holds *graph f is open Subset of [:X,Y:]

then S6[Y] by Lm8;
hence for X being non empty TopSpace
for f being continuous Function of X,(Sigma (InclPoset the topology of Y)) holds *graph f is open Subset of [:X,Y:] by Lm6; :: thesis: verum
end;
assume A1: for X being non empty TopSpace
for f being continuous Function of X,(Sigma (InclPoset the topology of Y)) holds *graph f is open Subset of [:X,Y:] ; :: thesis: InclPoset the topology of Y is continuous
S4[Y]
proof
let X be non empty TopSpace; :: thesis: for T being Scott TopAugmentation of InclPoset the topology of Y
for f being continuous Function of X,T holds *graph f is open Subset of [:X,Y:]

let T be Scott TopAugmentation of InclPoset the topology of Y; :: thesis: for f being continuous Function of X,T holds *graph f is open Subset of [:X,Y:]
let f be continuous Function of X,T; :: thesis: *graph f is open Subset of [:X,Y:]
A2: ( RelStr(# the carrier of T, the InternalRel of T #) = InclPoset the topology of Y & RelStr(# the carrier of (Sigma (InclPoset the topology of Y)), the InternalRel of (Sigma (InclPoset the topology of Y)) #) = InclPoset the topology of Y ) by YELLOW_9:def 4;
then reconsider g = f as Function of X,(Sigma (InclPoset the topology of Y)) ;
( TopStruct(# the carrier of X, the topology of X #) = TopStruct(# the carrier of X, the topology of X #) & TopStruct(# the carrier of T, the topology of T #) = TopStruct(# the carrier of (Sigma (InclPoset the topology of Y)), the topology of (Sigma (InclPoset the topology of Y)) #) ) by ;
then g is continuous by YELLOW12:36;
hence *graph f is open Subset of [:X,Y:] by A1; :: thesis: verum
end;
then S5[Y] by Lm4;
then S6[Y] by Lm5;
hence InclPoset the topology of Y is continuous by Lm7; :: thesis: verum