reserve T for TopSpace;
reserve A,B for Subset of T;
reserve T for non empty TopSpace;
reserve P,Q for Element of Topology_of T;
reserve p,q for Element of Open_setLatt(T);
reserve L for D_Lattice;
reserve F for Filter of L;
reserve a,b for Element of L;
reserve x,X,X1,X2,Y,Z for set;

theorem Th15:
  StoneH(L).(a "/\" b) = StoneH(L).a /\ StoneH(L).b
proof
  set c = a "/\" b;
  hereby
    set c = a "/\" b;
    let x be object;
    assume x in StoneH(L).c;
    then consider F such that
A1: x=F and
A2: F <> the carrier of L and
A3: F is prime and
A4: c in F by Th12;
    b in F by A4,FILTER_0:8;
    then
A5: F in StoneH(L).b by A2,A3,Th12;
    a in F by A4,FILTER_0:8;
    then F in StoneH(L).a by A2,A3,Th12;
    hence x in StoneH(L).a /\ StoneH(L).b by A1,A5,XBOOLE_0:def 4;
  end;
  let x be object;
  assume
A6: x in StoneH(L).a /\ StoneH(L).b;
  then x in StoneH(L).b by XBOOLE_0:def 4;
  then
A7: ex F st x=F & F <> the carrier of L & F is prime & b in F by Th12;
  x in StoneH(L).a by A6,XBOOLE_0:def 4;
  then ex F st x=F & F <> the carrier of L & F is prime & a in F by Th12;
  then consider F such that
A8: x=F and
A9: F <> the carrier of L and
A10: F is prime and
A11: a in F and
A12: b in F by A7;
  c in F by A11,A12,FILTER_0:8;
  hence thesis by A8,A9,A10,Th12;
end;
