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 Th14:
  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;
    a in F or b in F by A3,A4;
    then F in StoneH(L).a or F in StoneH(L).b by A2,A3,Th12;
    hence x in StoneH(L).a \/ StoneH(L).b by A1,XBOOLE_0:def 3;
  end;
  let x be object;
  assume x in StoneH(L).a \/ StoneH(L).b;
  then x in StoneH(L).a or x in StoneH(L).b by XBOOLE_0:def 3;
  then
  (ex F st x=F & F <> the carrier of L & F is prime & a in F ) or ex F st
  x=F & F <> the carrier of L & F is prime & b in F by Th12;
  then consider F such that
A5: x=F and
A6: F <> the carrier of L and
A7: F is prime and
A8: a in F or b in F;
  c in F by A7,A8;
  hence thesis by A5,A6,A7,Th12;
end;
