
theorem Th8:
  for L be lower-bounded sup-Semilattice holds InclPoset Ids L is
  CLSubFrame of BoolePoset the carrier of L
proof
  let L be lower-bounded sup-Semilattice;
  now
    let x be object;
    assume x in Ids L;
    then x in the set of all  X where X is Ideal of L by WAYBEL_0:def 23;
    then ex x9 be Ideal of L st x9 = x;
    hence x in bool the carrier of L;
  end;
  then Ids L is Subset-Family of L by TARSKI:def 3;
  then reconsider
  InIdL = InclPoset Ids L as non empty full SubRelStr of BoolePoset
  the carrier of L by YELLOW_1:5;
A1: for X be directed Subset of InIdL st X <> {} & ex_sup_of X,BoolePoset
  the carrier of L holds "\/"(X,BoolePoset the carrier of L) in the carrier of
  InIdL
  proof
    let X be directed Subset of InIdL;
    assume that
A2: X <> {} and
    ex_sup_of X,BoolePoset the carrier of L;
    consider Y be object such that
A3: Y in X by A2,XBOOLE_0:def 1;
    X is Subset of BoolePoset the carrier of L by Th3;
    then
A4: "\/"(X,BoolePoset the carrier of L) = union X by YELLOW_1:21;
    then reconsider uX = union X as Subset of L by WAYBEL_8:26;
    reconsider Y as set by TARSKI:1;
    Y is Ideal of L by A3,YELLOW_2:41;
    then Bottom L in Y by WAYBEL_4:21;
    then reconsider uX as non empty Subset of L by A3,TARSKI:def 4;
    now
      let z be object;
      assume z in X;
      then z is Ideal of L by YELLOW_2:41;
      hence z in bool the carrier of L;
    end;
    then
A5: X c= bool the carrier of L;
A6: now
      let Y,Z be Subset of L;
      assume
A7:   Y in X & Z in X;
      then reconsider Y9 = Y, Z9 = Z as Element of InIdL;
      consider V9 be Element of InIdL such that
A8:   V9 in X and
A9:   Y9 <= V9 & Z9 <= V9 by A7,WAYBEL_0:def 1;
      reconsider V = V9 as Subset of L by YELLOW_2:41;
      take V;
      thus V in X by A8;
      Y9 "\/" Z9 <= V9 by A9,YELLOW_0:22;
      then
A10:  Y9 "\/" Z9 c= V9 by YELLOW_1:3;
      Y \/ Z c= Y9 "\/" Z9 by YELLOW_1:6;
      hence Y \/ Z c= V by A10;
    end;
    ( for Y be Subset of L st Y in X holds Y is lower)& for Y be Subset
    of L st Y in X holds Y is directed by YELLOW_2:41;
    then uX is Ideal of L by A5,A6,WAYBEL_0:26,46;
    then "\/"(X,BoolePoset the carrier of L) is Element of InIdL by A4,
YELLOW_2:41;
    hence thesis;
  end;
  for X be Subset of InIdL st ex_inf_of X,BoolePoset the carrier of L
  holds "/\"(X,BoolePoset the carrier of L) in the carrier of InIdL
  proof
    let X be Subset of InIdL;
    assume ex_inf_of X,BoolePoset the carrier of L;
    now
      per cases;
      suppose
A11:    X is non empty;
        X is Subset of BoolePoset the carrier of L by Th3;
        then
A12:    "/\"(X,BoolePoset the carrier of L) = meet X by A11,YELLOW_1:20;
        InclPoset Ids L = RelStr (# Ids L, RelIncl Ids L #) by YELLOW_1:def 1;
        then "/\"(X,BoolePoset the carrier of L) is Ideal of L by A11,A12,
YELLOW_2:45;
        then "/\"(X,BoolePoset the carrier of L) is Element of InIdL by
YELLOW_2:41;
        hence thesis;
      end;
      suppose
A13:    X is empty;
        "/\"({},BoolePoset the carrier of L) = Top (BoolePoset the
        carrier of L ) by YELLOW_0:def 12
          .= the carrier of L by YELLOW_1:19;
        then "/\"({},BoolePoset the carrier of L) is Ideal of L by Th4;
        then "/\"(X,BoolePoset the carrier of L) is Element of InIdL by A13,
YELLOW_2:41;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
  hence thesis by A1,WAYBEL_0:def 4,YELLOW_0:def 18;
end;
