
theorem Th6:
  for L be lower-bounded sup-Semilattice for X be Subset of
  InclPoset Ids L holds sup X = downarrow finsups union X
proof
  let L be lower-bounded sup-Semilattice;
  let X be Subset of InclPoset Ids L;
  reconsider a = downarrow finsups union X as Element of InclPoset Ids L by
YELLOW_2:41;
A1: union X c= sup X by Lm1,YELLOW_0:17;
A2: now
    let b be Element of InclPoset Ids L;
    reconsider b1 = b as Ideal of L by YELLOW_2:41;
    assume b is_>=_than X;
    then b >= sup X by YELLOW_0:32;
    then sup X c= b1 by YELLOW_1:3;
    then union X c= b1 by A1;
    then downarrow finsups union X c= b1 by WAYBEL_0:61;
    hence a <= b by YELLOW_1:3;
  end;
A3: union X c= downarrow finsups union X by WAYBEL_0:61;
  now
    let b be Element of InclPoset Ids L;
    assume b in X;
    then b c= union X by ZFMISC_1:74;
    then b c= downarrow finsups union X by A3;
    hence b <= a by YELLOW_1:3;
  end;
  then a is_>=_than X;
  hence thesis by A2,YELLOW_0:32;
end;
