
theorem Th52:
  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)
  for x being Element of L holds x is_>=_than X iff x is_>=_than F
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;
  let x be Element of L;
  thus x is_>=_than X implies x is_>=_than F
  proof
    assume
A4: x is_>=_than X;
    let y be Element of L;
    assume y in F;
    then consider Y being finite Subset of X such that
A5: ex_sup_of Y,L and
A6: y = "\/"(Y,L) by A2;
    x is_>=_than Y by A4;
    hence thesis by A5,A6,YELLOW_0:def 9;
  end;
  assume
A7: x is_>=_than F;
  let y be Element of L;
  assume y in X;
  then
A8: {y} c= X by ZFMISC_1:31;
  then
A9: sup {y} in F by A3;
  ex_sup_of {y},L by A1,A8;
  then
A10: {y} is_<=_than sup {y} by YELLOW_0:def 9;
A11: sup {y} <= x by A7,A9;
  y <= sup {y} by A10,YELLOW_0:7;
  hence thesis by A11,ORDERS_2:3;
end;
