reserve x,y,X,X1,Y,Z for set;
reserve L for Lattice;
reserve F,H for Filter of L;
reserve p,q,r for Element of L;

theorem
  p in <.<.q.) \/ F.) implies ex r st r in F & q "/\" r [= p
proof
A1: <.<.q.) \/ F.)={r : ex p9,q9 being Element of L st p9"/\"q9 [= r & p9 in
  <.q.) & q9 in F} by FILTER_0:35;
  assume p in <.<.q.) \/ F.);
  then
  ex r st r=p & ex p9,q9 being Element of L st p9"/\"q9 [= r & p9 in <.q.)
  & q9 in F by A1;
  then consider p9,q9 being Element of L such that
A2: p9 "/\" q9 [= p and
A3: p9 in <.q.) and
A4: q9 in F;
  q [= p9 by A3,FILTER_0:15;
  then q "/\" q9 [= p9 "/\" q9 by LATTICES:9;
  hence thesis by A2,A4,LATTICES:7;
end;
