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 Th17:
  x in UFilter BL.a iff ex F st F=x & F is being_ultrafilter & a in F
proof
A1: x in UFilter BL.a implies ex F st F=x & F is being_ultrafilter & a in F
  proof
    assume x in UFilter BL.a;
    then x in {UF: UF is being_ultrafilter & a in UF} by Def6;
    then consider F such that
A2: F=x and
A3: F is being_ultrafilter and
A4: a in F;
    take F;
    thus thesis by A2,A3,A4;
  end;
  (ex F st F=x & F is being_ultrafilter & a in F) implies x in UFilter BL.a
  proof
    assume ex F st F=x & F is being_ultrafilter & a in F;
    then x in {UF: UF is being_ultrafilter & a in UF};
    hence thesis by Def6;
  end;
  hence thesis by A1;
end;
