
theorem Th30:
  for L being Semilattice, F being Filter of L, G being
GeneratorSet of F for A being non empty Subset of L st G is_coarser_than A & A
  is_coarser_than F holds A is GeneratorSet of F
proof
  let L be Semilattice, F be Filter of L, G be GeneratorSet of F, A be non
  empty Subset of L such that
A1: G is_coarser_than A;
  assume
A2: A is_coarser_than F;
  F = uparrow fininfs G by Def3;
  hence F c= uparrow fininfs A by A1,Th4,Th29;
  A c= F by A2,YELLOW_4:8;
  hence thesis by WAYBEL_0:62;
end;
