
theorem
  for Y being T_0-TopSpace holds InclPoset the topology of Y is
  continuous iff for X being non empty TopSpace holds Theta(X, Y) is isomorphic
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 a4102[Y] by Lm3;
  end;
  assume a4102[Y];
  then a4103[Y] by Lm2;
  then a4104[Y] by Lm4;
  then a4105[Y] by Lm5;
  hence thesis by Lm7;
end;
