
theorem Th1:
  for L being non empty transitive RelStr, X being Subset of L holds
  X is non empty directed iff for Y being finite Subset of X
  ex x being Element of L st x in X & x is_>=_than Y
proof
  let L be non empty transitive RelStr, X be Subset of L;
  hereby
    assume X is non empty;
    then reconsider X9 = X as non empty set;
    assume
A1: X is directed;
    let Y be finite Subset of X;
    defpred P[set] means ex x being Element of L st x in X & x is_>=_than $1;
A2: Y is finite;
    set x = the Element of X9;
    x in X;
    then reconsider x as Element of L;
    x is_>=_than {};
    then
A3: P[{}];
A4: now
      let x,B be set;
      assume that
A5:   x in Y and B c= Y;
      assume P[B];
      then consider y being Element of L such that
A6:   y in X and
A7:   y is_>=_than B;
      x in X by A5;
      then reconsider x9 = x as Element of L;
      consider z being Element of L such that
A8:   z in X and
A9:   x9 <= z and
A10:  y <= z by A1,A5,A6;
      thus P[B \/ {x}]
      proof
        take z;
        thus z in X by A8;
        let a be Element of L;
        reconsider a9 = a as Element of L;
        assume a in B \/ {x};
        then a9 in B or a9 in {x} by XBOOLE_0:def 3;
        then y >= a9 or a9 = x by A7,TARSKI:def 1;
        hence thesis by A9,A10,ORDERS_2:3;
      end;
    end;
    thus P[Y] from FINSET_1:sch 2(A2,A3,A4);
  end;
  assume
A11: for Y being finite Subset of X
  ex x being Element of L st x in X & x is_>=_than Y;
  {} c= X;
  then ex x being Element of L st x in X & x is_>=_than {} by A11;
  hence X is non empty;
  let x,y be Element of L;
  assume that
A12: x in X and
A13: y in X;
  {x,y} c= X by A12,A13,ZFMISC_1:32;
  then consider z being Element of L such that
A14: z in X and
A15: z is_>=_than {x,y} by A11;
  take z;
A16: x in {x,y} by TARSKI:def 2;
  y in {x,y} by TARSKI:def 2;
  hence thesis by A14,A15,A16;
end;
