reserve L for Lattice,
  p,p1,q,q1,r,r1 for Element of L;
reserve x,y,z,X,Y,Z,X1,X2 for set;
reserve H,F for Filter of L;

theorem Th20:
  L is 0_Lattice & p <> Bottom L implies ex H st p in H & H is
  being_ultrafilter
proof
  assume that
A1: L is 0_Lattice and
A2: p <> Bottom L;
  reconsider L9 = L as 0_Lattice by A1;
  reconsider p9 = p as Element of L9;
  p9 "/\" Bottom L9 = Bottom L9;
  then consider H such that
A3: <.p.) c= H & H is being_ultrafilter by A2,Th18,Th19;
  take H;
  p in <.p.);
  hence thesis by A3;
end;
