
theorem
  for L being with_suprema Poset, X being Subset of L
  st ex_sup_of X,L or L is complete holds sup X = sup finsups X
proof
  let L be with_suprema Poset, X be Subset of L;
  assume ex_sup_of X,L or L is complete;
  then
A1: ex_sup_of X,L by YELLOW_0:17;
A2: now
    let Y be finite Subset of X;
    Y c= the carrier of L by XBOOLE_1:1;
    hence Y <> {} implies ex_sup_of Y,L by YELLOW_0:54;
  end;
A3: now
    let x be Element of L;
    assume x in finsups X;
    then ex Y being finite Subset of X st x = "\/"(Y,L) & ex_sup_of Y,L;
    hence ex Y being finite Subset of X st ex_sup_of Y,L & x = "\/"(Y,L);
  end;
  now
    let Y be finite Subset of X;
    reconsider Z = Y as Subset of L by XBOOLE_1:1;
    assume Y <> {};
    then ex_sup_of Z,L by YELLOW_0:54;
    hence "\/"(Y,L) in finsups X;
  end;
  hence thesis by A1,A2,A3,Th54;
end;
