
theorem
  for L being with_suprema Poset, X being Subset of L holds
  X c= downarrow finsups X &
  for I being Ideal of L st X c= I holds downarrow finsups X c= I
proof
  let L be with_suprema Poset, X be Subset of L;
A1: X c= finsups X by Th50;
  finsups X c= downarrow finsups X by Th16;
  hence X c= downarrow finsups X by A1;
  let I be Ideal of L such that
A2: X c= I;
  let x be object;
  assume
A3: x in downarrow finsups X;
  then reconsider x as Element of L;
  consider y being Element of L such that
A4: x <= y and
A5: y in finsups X by A3,Def15;
  consider Y being finite Subset of X such that
A6: y = "\/"(Y,L) and
A7: ex_sup_of Y,L by A5;
  set i = the Element of I;
  reconsider i as Element of L;
A8: ex_sup_of {i}, L by YELLOW_0:38;
A9: sup {i} = i by YELLOW_0:39;
A10: now
    assume ex_sup_of {},L;
    then "\/"({},L) <= sup {i} by A8,XBOOLE_1:2,YELLOW_0:34;
    hence "\/"({},L) in I by A9,Def19;
  end;
  Y c= I by A2;
  then y in I by A6,A7,A10,Th42;
  hence thesis by A4,Def19;
end;
