reserve X,Y,x,y for set;
reserve A for non empty Poset;
reserve a,a1,a2,a3,b,c for Element of A;
reserve S,T for Subset of A;
reserve f for Choice_Function of BOOL(the carrier of A);
reserve fC,fC1,fC2 for Chain of f;
reserve R for Relation,
  A for non empty Poset,
  C for Chain of A,
  S for Subset of A,
  a,a1,a2,b,c1,c2 for Element of A;

theorem
  (for C ex a st for b st b in C holds a <= b) implies ex a st for b
  holds not b < a
proof
  set X = the carrier of A;
  set R = (the InternalRel of A)~;
A1: dom R = dom the InternalRel of A by RELAT_2:12
    .= X by PARTFUN1:def 2;
A2: for a,b being Element of A holds [a,b] in R iff b <= a
  by RELAT_1:def 7;
  reconsider R as Order of the carrier of A by A1,PARTFUN1:def 2;
  set B = RelStr (# the carrier of A, R #);
  assume
A3: for C ex a st for b st b in C holds a <= b;
  for E being Chain of B ex e being Element of B st for f being Element of
  B st f in E holds f <= e
  proof
    let E be Chain of B;
    reconsider C = E as Subset of A;
    the InternalRel of A is_strongly_connected_in C
    proof
      let x,y be object;
      assume
A4:   x in C & y in C;
      then reconsider e = x, f = y as Element of B;
      reconsider e1 = e, f1 = f as Element of A;
A5:   e <= f or f <= e by A4,Th11;
      now
        per cases by A5;
        suppose
          [e,f] in R;
          then f1 <= e1 by A2;
          hence thesis;
        end;
        suppose
          [f,e] in R;
          then e1 <= f1 by A2;
          hence thesis;
        end;
      end;
      hence thesis;
    end;
    then reconsider C as Chain of A by Def7;
    consider a such that
A6: for b st b in C holds a <= b by A3;
    reconsider e = a as Element of B;
    take e;
    let f be Element of B;
    reconsider b = f as Element of A;
    assume f in E;
    then a <= b by A6;
    then [f,e] in R by A2;
    hence thesis;
  end;
  then consider e being Element of B such that
A7: for f being Element of B holds not e < f by Th55;
  reconsider d = e as Element of A;
  take d;
  let b;
  reconsider f = b as Element of B;
  assume
A8: b < d;
  then b <= d;
  then [e,f] in R by A2;
  then e <= f;
  then e < f by A8;
  hence thesis by A7;
end;
