
theorem Th6:
  for L being upper-bounded with_infima non empty Poset holds
  InclPoset Filt L is CLSubFrame of BoolePoset the carrier of L
proof
  let L be upper-bounded with_infima non empty Poset;
  set cL = the carrier of L;
  set BP = BoolePoset cL;
  set cBP = the carrier of BP;
  set rBP = the InternalRel of BP;
  set IP = InclPoset Filt L;
  set cIP = the carrier of IP;
  set rIP = the InternalRel of IP;
A1: InclPoset bool cL = RelStr(#bool cL, RelIncl bool cL#) by YELLOW_1:def 1;
A2: InclPoset Filt L = RelStr(#Filt L, RelIncl Filt L#) by YELLOW_1:def 1;
A3: BoolePoset cL = InclPoset bool cL by YELLOW_1:4;
A4: cIP c= cBP
  proof
    let x be object;
    assume x in cIP;
    then ex X being Filter of L st x = X by A2;
    hence thesis by A3,A1;
  end;
A5: field rIP = Filt L by A2,WELLORD2:def 1;
  rIP c= rBP
  proof
    let p be object;
    assume
A6: p in rIP;
    then consider x, y being object such that
A7: p = [x, y] by RELAT_1:def 1;
A8: y in field rIP by A6,A7,RELAT_1:15;
    then consider Y being Filter of L such that
A9: y = Y by A5;
A10: x in field rIP by A6,A7,RELAT_1:15;
    then consider X being Filter of L such that
A11: x = X by A5;
    X c= Y by A2,A5,A6,A7,A10,A8,A11,A9,WELLORD2:def 1;
    hence thesis by A3,A1,A7,A11,A9,WELLORD2:def 1;
  end;
  then reconsider IP as SubRelStr of BP by A4,YELLOW_0:def 13;
  now
    let p be object;
A12: rBP|_2 cIP = rBP /\ [:cIP, cIP:] by WELLORD1:def 6;
    hereby
      assume
A13:  p in rIP;
      then consider x, y being object such that
A14:  p = [x, y] by RELAT_1:def 1;
A15:  y in field rIP by A13,A14,RELAT_1:15;
      then consider Y being Filter of L such that
A16:  y = Y by A5;
A17:  x in field rIP by A13,A14,RELAT_1:15;
      then consider X being Filter of L such that
A18:  x = X by A5;
      X c= Y by A2,A5,A13,A14,A17,A15,A18,A16,WELLORD2:def 1;
      then p in rBP by A3,A1,A14,A18,A16,WELLORD2:def 1;
      hence p in rBP|_2 cIP by A12,A13,XBOOLE_0:def 4;
    end;
    assume
A19: p in rBP|_2 cIP;
    then
A20: p in rBP by A12,XBOOLE_0:def 4;
    p in [:cIP, cIP:] by A12,A19,XBOOLE_0:def 4;
    then consider x, y being object such that
A21: x in cIP & y in cIP and
A22: p = [x, y] by ZFMISC_1:def 2;
    reconsider x,y as set by TARSKI:1;
    (ex X being Filter of L st x = X )& ex Y being Filter of L st y = Y
    by A2,A21;
    then x c= y by A3,A1,A20,A22,WELLORD2:def 1;
    hence p in rIP by A2,A21,A22,WELLORD2:def 1;
  end;
  then rIP = rBP|_2 cIP by TARSKI:2;
  then reconsider IP as full SubRelStr of BP by YELLOW_0:def 14;
A23: Filt L c= bool cL
  proof
    let x be object;
    assume x in Filt L;
    then ex X being Filter of L st x = X;
    hence thesis;
  end;
A24: IP is directed-sups-inheriting
  proof
    let X be directed Subset of IP such that
A25: X <> {} and
    ex_sup_of X, BP;
    consider Y being object such that
A26: Y in X by A25,XBOOLE_0:def 1;
    reconsider F = X as Subset-Family of cL by A2,A23,XBOOLE_1:1;
A27: for P, R being Subset of L st P in F & R in F ex Z being Subset of L
    st Z in F & P \/ R c= Z
    proof
      let P, R be Subset of L;
      assume
A28:  P in F & R in F;
      then reconsider P9 = P, R9 = R as Element of IP;
      consider Z being Element of IP such that
A29:  Z in X and
A30:  P9 <= Z & R9 <= Z by A28,WAYBEL_0:def 1;
      Z in the carrier of IP by A29;
      then consider Z9 being Filter of L such that
A31:  Z9 = Z by A2;
      take Z9;
      thus Z9 in F by A29,A31;
      P9 c= Z & R9 c= Z by A30,YELLOW_1:3;
      hence thesis by A31,XBOOLE_1:8;
    end;
    reconsider X9 = X as Subset of BP by A3,A1,A2,A23,XBOOLE_1:1;
    set sX = "\/"(X, BP);
A32: sup X9 = union X by YELLOW_1:21;
    reconsider sX as Subset of L by A1,YELLOW_1:4;
A33: for X being Subset of L st X in F holds X is upper filtered
    proof
      let X be Subset of L;
      assume X in F;
      then X in Filt L by A2;
      then ex Y being Filter of L st X = Y;
      hence thesis;
    end;
    then for X being Subset of L st X in F holds X is upper;
    then
A34: sX is upper by A32,WAYBEL_0:28;
    for X being Subset of L st X in F holds X is filtered by A33;
    then
A35: sX is filtered by A32,A27,WAYBEL_0:47;
    reconsider Y as set by TARSKI:1;
    Y in Filt L by A2,A26;
    then ex Z being Filter of L st Y = Z;
    then Top L in Y by WAYBEL_4:22;
    then sX is non empty by A32,A26,TARSKI:def 4;
    hence thesis by A2,A34,A35;
  end;
  IP is infs-inheriting
  proof
    let X be Subset of IP such that
    ex_inf_of X, BP;
    set sX = "/\"(X, BP);
    per cases;
    suppose
A36:  X is empty;
A37:  [#]L = cL;
      "/\"(X, BP) = Top BP by A36
        .= cL by YELLOW_1:19;
      hence thesis by A2,A37;
    end;
    suppose
A38:  X is non empty;
      reconsider F = X as Subset-Family of cL by A2,A23,XBOOLE_1:1;
      reconsider sX as Subset of L by A1,YELLOW_1:4;
      reconsider X9 = X as Subset of BP by A3,A1,A2,A23,XBOOLE_1:1;
A39:  inf X9 = meet X by A38,YELLOW_1:20;
A40:  for X being Subset of L st X in F holds X is upper filtered
      proof
        let X be Subset of L;
        assume X in F;
        then X in Filt L by A2;
        then ex Y being Filter of L st X = Y;
        hence thesis;
      end;
      then
A41:  sX is filtered by A39,YELLOW_2:39;
      for X being Subset of L st X in F holds X is upper by A40;
      then
A42:  sX is upper by A39,YELLOW_2:37;
      for Y being set st Y in X holds Top L in Y
      proof
        let Y be set;
        assume Y in X;
        then Y in Filt L by A2;
        then ex Z being Filter of L st Y = Z;
        hence thesis by WAYBEL_4:22;
      end;
      then sX is non empty by A38,A39,SETFAM_1:def 1;
      hence thesis by A2,A42,A41;
    end;
  end;
  hence thesis by A24;
end;
