reserve x,y,X for set;

theorem Th15:
  for T being non empty 1-sorted for F being Filter of BoolePoset
  [#]T for A being non empty Subset of T holds A in F iff a_net F
  is_eventually_in A
proof
  let T be non empty 1-sorted;
  let F be Filter of BoolePoset [#]T;
  let B be non empty Subset of T;
A1: B in F iff B in F\{{}} by ZFMISC_1:56;
  F\{{}} = a_filter a_net F by Th13;
  hence thesis by A1,Th10;
end;
