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 Th20:
  UFilter BL.(a "/\" b) = UFilter BL.a /\ UFilter BL.b
proof
A1: UFilter BL.(a "/\" b) c= UFilter BL.a /\ UFilter BL.b
  proof
    let x be object;
    set c = a "/\" b;
    assume x in UFilter BL.c;
    then consider F0 such that
A2: x=F0 and
A3: F0 is being_ultrafilter and
A4: c in F0 by Th17;
A5: a in F0 by A4,FILTER_0:8;
A6: b in F0 by A4,FILTER_0:8;
A7: F0 in UFilter BL.(a) by A3,A5,Th17;
    F0 in UFilter BL.(b) by A3,A6,Th17;
    hence thesis by A2,A7,XBOOLE_0:def 4;
  end;
  UFilter BL.a /\ UFilter BL.b c= UFilter BL.(a "/\" b)
  proof
    let x be object;
    assume
A8: x in UFilter BL.a /\ UFilter BL.b;
    then
A9: x in UFilter BL.a by XBOOLE_0:def 4;
A10: x in UFilter BL.b by A8,XBOOLE_0:def 4;
A11: ex F0 st x=F0 & F0 is being_ultrafilter & a in F0 by A9,Th17;
    ex F0 st x=F0 & F0 is being_ultrafilter & b in F0 by A10,Th17;
    then consider F0 such that
A12: x=F0 and
A13: F0 is being_ultrafilter and
A14: a in F0 and
A15: b in F0 by A11;
    a "/\" b in F0 by A14,A15,FILTER_0:8;
    hence thesis by A12,A13,Th17;
  end;
  hence thesis by A1;
end;
