reserve x,y,X for set;

theorem Th13:
  for T being non empty 1-sorted for F being Filter of BoolePoset
  [#]T holds F \ {{}} = a_filter a_net F
proof
  let T be non empty 1-sorted;
  let F be Filter of BoolePoset [#]T;
  set X = a_filter a_net F;
A1: the carrier of a_net F = {[a, f] where a is Element of T, f is Element
  of F: a in f} by Def4;
A2: BoolePoset [#]T = InclPoset bool [#]T by YELLOW_1:4;
  thus F \ {{}} c= X
  proof
    let x be object;
    assume
A3: x in F \ {{}};
    then reconsider A = x as Subset of T by A2,YELLOW_1:1;
    set a = the Element of A;
    not x in {{}} by A3,XBOOLE_0:def 5;
    then
A4: A <> {} by TARSKI:def 1;
    then a in A;
    then reconsider a as Element of T;
    x in F by A3,XBOOLE_0:def 5;
    then [a, A] in the carrier of a_net F by A1,A4;
    then reconsider i = [a, A] as Element of a_net F;
    a_net F is_eventually_in A
    proof
      take i;
      let j be Element of a_net F;
A5:   (a_net F).j = j`1 by Def4;
      assume i <= j;
      then
A6:   j`2 c= i`2 by Def4;
      j in the carrier of a_net F;
      then consider a being Element of T, f being Element of F such that
A7:   j = [a,f] and
A8:   a in f by A1;
A9:  j`1 = a by A7;
      j`2 = f by A7;
      hence thesis by A8,A6,A5,A9;
    end;
    hence thesis;
  end;
  let x be object;
   reconsider xx=x as set by TARSKI:1;
  assume
A10: x in X;
  then a_net F is_eventually_in xx by Th10;
  then consider i being Element of a_net F such that
A11: for j being Element of a_net F st i <= j holds (a_net F).j in xx;
  i in the carrier of a_net F;
  then consider a being Element of T, f being Element of F such that
A12: i = [a,f] and
A13: a in f by A1;
A14: the carrier of BoolePoset [#]T = bool [#]T by A2,YELLOW_1:1;
A15: f c= xx
  proof
    let x be object;
    assume
A16: x in f;
    then reconsider b = x as Element of T by A14;
    [b,f] in the carrier of a_net F by A1,A16;
    then reconsider j = [b,f] as Element of a_net F;
A17: j`2 = f;
    j`1 = b;
    then
A18: (a_net F).j = b by Def4;
    i`2 = f by A12;
    then i <= j by A17,Def4;
    hence thesis by A11,A18;
  end;
  x is Subset of T by A10,Th10;
  then
A19: x in F by A15,WAYBEL_7:7;
  not x in {{}} by A13,A15,TARSKI:def 1;
  hence thesis by A19,XBOOLE_0:def 5;
end;
