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 Th12:
  x in StoneH(L).a iff ex F st F=x & F <> the carrier of L & F is
  prime & a in F
proof
A1: StoneH(L).a = {F: F in F_primeSet(L) & a in F} by Def6;
  hereby
    assume x in StoneH(L).a;
    then consider F such that
A2: x=F and
A3: F in F_primeSet(L) and
A4: a in F by A1;
A5: F is prime by A3,Th10;
    F <> the carrier of L by A3,Th10;
    hence ex F st F=x & F <> the carrier of L & F is prime & a in F by A2,A4,A5
;
  end;
  given F such that
A6: F=x and
A7: F <> the carrier of L and
A8: F is prime and
A9: a in F;
  F in F_primeSet(L) by A7,A8;
  hence thesis by A1,A6,A9;
end;
