
theorem Th51:
  for L being non empty transitive RelStr for 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)
  holds F is directed
proof
  let L be non empty transitive 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,y be Element of L;
  assume
A4: x in F;
  then consider Y1 being finite Subset of X such that
A5: ex_sup_of Y1,L and
A6: x = "\/"(Y1,L) by A2;
  assume y in F;
  then consider Y2 being finite Subset of X such that
A7: ex_sup_of Y2,L and
A8: y = "\/"(Y2,L) by A2;
  take z = "\/"(Y1 \/ Y2, L);
A9: Y1 = {} & Y2 = {} & {} \/ {} = {} or Y1 \/ Y2 <> {};
  hence z in F by A3,A4,A6;
  ex_sup_of Y1 \/ Y2,L by A1,A5,A9;
  hence thesis by A5,A6,A7,A8,XBOOLE_1:7,YELLOW_0:34;
end;
