theorem Th4:
  { X } is Filter of X
proof
A1: for Y1,Y2 holds (Y1 in { X } & Y2 in { X } implies Y1 /\ Y2 in { X }) &
  ( Y1 in { X } & Y1 c= Y2 implies Y2 in { X }) by Th2;
  { X } is non empty Subset-Family of X & not {} in { X } by Th2;
  hence thesis by A1,Def1;
end;
