
theorem
  for Y being T_0-TopSpace holds 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
  :]
proof
  let Y be T_0-TopSpace;
  hereby
    assume InclPoset the topology of Y is continuous;
    then a4105[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;
  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:];
  a4103[Y]
  proof
    let X be non empty TopSpace;
    let T be Scott TopAugmentation of InclPoset the topology of Y;
    let f be continuous Function of X, T;
A2: the RelStr of T = InclPoset the topology of Y & the RelStr 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;
    the TopStruct of X = the TopStruct of X & the TopStruct of T = the
    TopStruct of Sigma InclPoset the topology of Y by A2,Th13;
    then g is continuous by YELLOW12:36;
    hence thesis by A1;
  end;
  then a4104[Y] by Lm4;
  then a4105[Y] by Lm5;
  hence thesis by Lm7;
end;
