reserve x,y,z,X for set,
  T for Universe;

theorem Th31:
  for T be non empty TopSpace, p be Point of T for x,y being
  Element of OpenNeighborhoods p holds x <= y iff y c= x
proof
  let T be non empty TopSpace, p be Point of T;
  set X = { V where V is Subset of T: p in V & V is open };
  [#]T in X;
  then reconsider X as non empty set;
  let x,y be Element of OpenNeighborhoods p;
  (InclPoset X)~ = RelStr(#the carrier of InclPoset X, (the InternalRel of
    InclPoset X)~#) by LATTICE3:def 5;
  then reconsider a = x, b = y as Element of InclPoset X;
A1: b <= a iff y c= x by YELLOW_1:3;
  x = a~ & y = b~ by LATTICE3:def 6;
  hence thesis by A1,LATTICE3:9;
end;
