reserve L for Lattice,
  p,p1,q,q1,r,r1 for Element of L;
reserve x,y,z,X,Y,Z,X1,X2 for set;

theorem Th9:
  for D being non empty Subset of L holds D is Filter of L iff (for
p,q st p in D & q in D holds p "/\" q in D) & for p,q st p in D & p [= q holds
  q in D
proof
  let D be non empty Subset of L;
  thus D is Filter of L implies (for p,q st p in D & q in D holds p "/\" q in
  D) & for p,q st p in D & p [= q holds q in D by LATTICES:def 23,def 24;
  assume
A1: ( for p,q st p in D & q in D holds p "/\" q in D)& for p,q st p in D
  & p [= q holds q in D;
  then for p,q st p [= q & p in D holds q in D;
 hence thesis by A1,LATTICES:def 23,def 24;
end;
