
theorem
  for Y being T_0-TopSpace holds InclPoset the topology of Y is
continuous iff for y being Element of Y, V being open a_neighborhood of y ex H
  being open Subset of Sigma InclPoset the topology of Y st V in H & meet H is
  a_neighborhood of y
proof
  let Y be T_0-TopSpace;
  thus InclPoset the topology of Y is continuous implies for y being Element
  of Y, V being open a_neighborhood of y ex H being open Subset of Sigma
  InclPoset the topology of Y st V in H & meet H is a_neighborhood of y by Lm8;
  assume
A1: for y being Element of Y, V being open a_neighborhood of y ex H
  being open Subset of Sigma InclPoset the topology of Y st V in H & meet H is
  a_neighborhood of y;
  a4105[Y]
  proof
    let T be Scott TopAugmentation of InclPoset the topology of Y;
    let y be Element of Y, V be open a_neighborhood of y;
    consider H being open Subset of Sigma InclPoset the topology of Y such
    that
A2: V in H & meet H is a_neighborhood of y by A1;
    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 = H as Subset of T;
    the topology of T = the topology of Sigma InclPoset the topology of Y
    by Th13;
    then G in the topology of T by PRE_TOPC:def 2;
    then H is open Subset of T by PRE_TOPC:def 2;
    hence thesis by A2;
  end;
  hence thesis by Lm7;
end;
