
theorem
  for Y being T_0-TopSpace holds InclPoset the topology of Y is
  continuous iff {[W,y] where W is open Subset of Y, y is Element of Y: y in W}
  is open Subset of [:Sigma InclPoset the topology of Y, Y:]
proof
  let Y be T_0-TopSpace;
  hereby
    assume InclPoset the topology of Y is continuous;
    then a4105[Y] by Lm8;
    then a4103[Y] by Lm6;
    hence {[W,y] where W is open Subset of Y, y is Element of Y: y in W} is
    open Subset of [:Sigma InclPoset the topology of Y, Y:] by Lm4;
  end;
  assume
A1: {[W,y] where W is open Subset of Y, y is Element of Y: y in W} is
  open Subset of [:Sigma InclPoset the topology of Y, Y:];
  a4104[Y]
  proof
    let T be Scott TopAugmentation of InclPoset the topology of Y;
    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 the TopStruct of Y = the TopStruct of Y & the TopStruct of T = the
    TopStruct of Sigma InclPoset the topology of Y by Th13;
    then [:T, Y:] = [:Sigma InclPoset the topology of Y, Y:] by Th7;
    hence thesis by A1;
  end;
  then a4105[Y] by Lm5;
  hence thesis by Lm7;
end;
