
theorem Th10: :: PROPOSITION 4.12.(i)
  for S be lower-bounded sup-Semilattice holds InclPoset Ids S is algebraic
proof
  let S be lower-bounded sup-Semilattice;
  set BS = BoolePoset (the carrier of S);
  Ids S c= bool the carrier of S
  proof
    let x be object;
    assume x in Ids S;
    then x in the set of all  X where X is Ideal of S  by
WAYBEL_0:def 23;
    then ex x1 be Ideal of S st x = x1;
    hence thesis;
  end;
  then reconsider InIdsS = InclPoset Ids S as non empty full SubRelStr of
  BoolePoset (the carrier of S) by YELLOW_1:5;
A1: the carrier of InIdsS c= the carrier of BS by YELLOW_0:def 13;
  now
    let X be Subset of InIdsS;
    assume ex_inf_of X,BS;
    now
      per cases;
      suppose
A2:     X <> {};
        for x being object st x in X holds x in the carrier of BS by A1;
        then X c= the carrier of BS;
        then "/\"(X,BS) = meet X by A2,YELLOW_1:20
          .= "/\"(X,InIdsS) by A2,YELLOW_2:46;
        hence "/\"(X,BS) in the carrier of InIdsS;
      end;
      suppose
A3:     X = {};
        "/\"({},BS) = Top BS by YELLOW_0:def 12
          .= the carrier of S by YELLOW_1:19
          .= [#]S
          .= "/\"({},InIdsS) by YELLOW_2:47;
        hence "/\"(X,BS) in the carrier of InIdsS by A3;
      end;
    end;
    hence "/\"(X,BS) in the carrier of InIdsS;
  end;
  then
A4: InIdsS is infs-inheriting by YELLOW_0:def 18;
  now
    let Y be directed Subset of InIdsS;
    assume that
A5: Y <> {} and
    ex_sup_of Y,BS;
    for x being object st x in Y holds x in the carrier of BS by A1;
    then Y c= the carrier of BS;
    then "\/"(Y,BS) = union Y by YELLOW_1:21
      .= "\/"(Y,InIdsS) by A5,Th9;
    hence "\/"(Y,BS) in the carrier of InIdsS;
  end;
  then InIdsS is directed-sups-inheriting by WAYBEL_0:def 4;
  hence thesis by A4,Th6;
end;
