
theorem Th9:
  for L be lower-bounded sup-Semilattice for X be non empty
  directed Subset of InclPoset Ids L holds sup X = union X
proof
  let L be lower-bounded sup-Semilattice;
  let X be non empty directed Subset of InclPoset Ids L;
  consider z be object such that
A1: z in X by XBOOLE_0:def 1;
  X c= the carrier of InclPoset Ids L;
  then
A2: X c= Ids L by YELLOW_1:1;
  now
    let x be object;
    assume x in X;
    then x in Ids L by A2;
    then x in the set of all  Y where Y is Ideal of L by WAYBEL_0:def 23;
    then ex x1 be Ideal of L st x = x1;
    hence x in bool the carrier of L;
  end;
  then
A3: X c= bool the carrier of L;
  now
    let Z be Subset of L;
    assume Z in X;
    then Z in Ids L by A2;
    then Z in the set of all  Y where Y is Ideal of L by WAYBEL_0:def 23;
    then ex Z1 be Ideal of L st Z = Z1;
    hence Z is lower;
  end;
  then reconsider unX = union X as lower Subset of L by A3,WAYBEL_0:26;
   reconsider z as set by TARSKI:1;
  z in Ids L by A2,A1;
  then z in the set of all  Y where Y is Ideal of L by WAYBEL_0:def 23;
  then ex z1 be Ideal of L st z = z1;
  then ex v be object st v in z by XBOOLE_0:def 1;
  then reconsider unX as lower non empty Subset of L by A1,TARSKI:def 4;
A4: now
    let V,Y be Subset of L;
    assume
A5: V in X & Y in X;
    then reconsider V1 = V, Y1 = Y as Element of InclPoset Ids L;
    consider Z1 be Element of InclPoset Ids L such that
A6: Z1 in X and
A7: V1 <= Z1 & Y1 <= Z1 by A5,WAYBEL_0:def 1;
    Z1 in Ids L by A2,A6;
    then Z1 in the set of all  Y9 where Y9 is Ideal of L by
WAYBEL_0:def 23;
    then ex Z2 be Ideal of L st Z1 = Z2;
    then reconsider Z = Z1 as Subset of L;
    take Z;
    thus Z in X by A6;
    V c= Z & Y c= Z by A7,YELLOW_1:3;
    hence V \/ Y c= Z by XBOOLE_1:8;
  end;
  now
    let Z be Subset of L;
    assume Z in X;
    then Z in Ids L by A2;
    then Z in the set of all  Y where Y is Ideal of L by WAYBEL_0:def 23;
    then ex Z1 be Ideal of L st Z = Z1;
    hence Z is directed;
  end;
  then reconsider unX as directed lower non empty Subset of L by A3,A4,
WAYBEL_0:46;
  reconsider unX as Element of InclPoset Ids L by YELLOW_2:41;
  now
    let Y be set;
    assume
A8: Y in X;
    then reconsider Y9 = Y as Element of InclPoset Ids L;
    sup X is_>=_than X by YELLOW_0:32;
    then Y9 <= sup X by A8,LATTICE3:def 9;
    hence Y c= sup X by YELLOW_1:3;
  end;
  then union X c= sup X by ZFMISC_1:76;
  then
A9: unX <= sup X by YELLOW_1:3;
  for b be Element of InclPoset Ids L st b in X holds b <= unX
         by YELLOW_1:3,ZFMISC_1:74;
  then unX is_>=_than X by LATTICE3:def 9;
  then sup X <= unX by YELLOW_0:32;
  hence thesis by A9,ORDERS_2:2;
end;
