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 Th25:
  ultraset BL \ UFilter BL.a = UFilter BL.a`
proof
  hereby
    let x be object;
    assume
A1: x in ultraset BL \ UFilter BL.a;
    then
A2: x in ultraset BL by XBOOLE_0:def 5;
A3: not x in UFilter BL.a by A1,XBOOLE_0:def 5;
    consider F such that
A4: F=x and
A5: F is being_ultrafilter by A2;
    not a in F by A3,A4,A5,Th18;
    then a` in F by A5,FILTER_0:46;
    hence x in UFilter BL.a` by A4,A5,Th18;
  end;
    let x be object;
    assume x in UFilter BL.a`;
    then consider F such that
A6: F=x and
A7: F is being_ultrafilter and
A8: a` in F by Th17;
A9: not a in F by A7,A8,FILTER_0:46;
A10: F in ultraset BL by A7;
    not F in UFilter BL.a by A9,Th18;
    hence x in ultraset BL \ UFilter BL.a by A6,A10,XBOOLE_0:def 5;
end;
