
theorem Th54:
  for L being non empty transitive reflexive RelStr, X,F being Subset of L st
  (for Y being finite Subset of X st Y <> {} holds ex_sup_of Y,L) &
  (for x being Element of L st x in F
  ex Y being finite Subset of X st ex_sup_of Y,L & x = "\/"(Y,L)) &
  (for Y being finite Subset of X st Y <> {} holds "\/"(Y,L) in F) &
  ex_sup_of X,L holds sup F = sup X
proof
  let L be non empty transitive reflexive RelStr;
  let X,F be Subset of L such that
A1: for Y being finite Subset of X st Y <> {} holds ex_sup_of Y,L and
A2: for x being Element of L st x in F
  ex Y being finite Subset of X st ex_sup_of Y,L & x = "\/"(Y,L) and
A3: for Y being finite Subset of X st Y <> {} holds "\/"(Y,L) in F;
  for x being Element of L holds x is_>=_than X iff x is_>=_than F
  by A1,A2,A3,Th52;
  hence thesis by YELLOW_0:47;
end;
