reserve T for non empty TopSpace,
  X,Z for Subset of T;
reserve x,y for Element of OpenClosedSet(T);
reserve x,y,X for set;
reserve BL for non trivial B_Lattice,
  a,b,c,p,q for Element of BL,
  UF,F,F0,F1,F2 for Filter of BL;

theorem Th30:
  UFilter BL.Bottom BL = {}
proof
  assume
A1: UFilter BL.Bottom BL <> {};
  set x = the Element of UFilter BL.Bottom BL;
  ex F st F=x & F is being_ultrafilter & Bottom BL in F by A1,Th17;
  hence contradiction by Th29;
end;
