
theorem Th22:
  for L being non empty reflexive transitive RelStr, R being
auxiliary(i) auxiliary(ii) (Relation of L), C being strict_chain of R st (for c
being Element of L holds ex_sup_of SetBelow (R,C,c),L) holds SupBelow (R,C) is
  strict_chain of R
proof
  let L be non empty reflexive transitive RelStr, R be auxiliary(i)
  auxiliary(ii) (Relation of L), C be strict_chain of R;
  assume
A1: for c being Element of L holds ex_sup_of SetBelow (R,C,c),L;
  thus SupBelow (R,C) is strict_chain of R
  proof
    let a, b be set;
    assume
A2: a in SupBelow (R,C);
    then
A3: a = sup SetBelow (R,C,a) by Def10;
    reconsider a as Element of L by A2;
A4: a <= a;
A5: ex_sup_of SetBelow (R,C,a),L by A1;
    assume
A6: b in SupBelow (R,C);
    then
A7: b = sup SetBelow (R,C,b) by Def10;
    reconsider b as Element of L by A6;
A8: b <= b;
A9: ex_sup_of SetBelow (R,C,b),L by A1;
    per cases;
    suppose
      a = b;
      hence thesis;
    end;
    suppose
A10:  a <> b;
      (for x being Element of L st x in C holds [x,a] in R iff [x,b] in R
      ) implies a = b
      proof
        assume
A11:    for x being Element of L st x in C holds [x,a] in R iff [x,b] in R;
        SetBelow (R,C,a) = SetBelow (R,C,b)
        proof
          thus SetBelow (R,C,a) c= SetBelow (R,C,b)
          proof
            let x be object;
            assume
A12:        x in SetBelow (R,C,a);
            then reconsider x as Element of L;
A13:        x in C by A12,Th15;
            [x,a] in R by A12,Th15;
            then [x,b] in R by A13,A11;
            hence thesis by A13,Th15;
          end;
          let x be object;
          assume
A14:      x in SetBelow (R,C,b);
          then reconsider x as Element of L;
A15:      x in C by A14,Th15;
          [x,b] in R by A14,Th15;
          then [x,a] in R by A15,A11;
          hence thesis by A15,Th15;
        end;
        hence thesis by A2,A7,Def10;
      end;
      then consider x being Element of L such that
A16:  x in C and
A17:  [x,a] in R & not [x,b] in R or not [x,a] in R & [x,b] in R by A10;
A18:  x <= x;
      thus thesis
      proof
        per cases by A17;
        suppose that
A19:      [x,a] in R and
A20:      not [x,b] in R;
          SetBelow (R,C,b) is_<=_than x
          proof
            let y be Element of L;
            assume
A21:        y in SetBelow (R,C,b);
            then [y,b] in R by Th15;
            then
A22:        y <= b by WAYBEL_4:def 3;
            y in C by A21,Th15;
            then [y,x] in R or x = y or [x,y] in R by A16,Def3;
            hence y <= x by A18,A20,A22,WAYBEL_4:def 3,def 4;
          end;
          then b <= x by A7,A9,YELLOW_0:def 9;
          hence thesis by A4,A19,WAYBEL_4:def 4;
        end;
        suppose that
A23:      not [x,a] in R and
A24:      [x,b] in R;
          SetBelow (R,C,a) is_<=_than x
          proof
            let y be Element of L;
            assume
A25:        y in SetBelow (R,C,a);
            then [y,a] in R by Th15;
            then
A26:        y <= a by WAYBEL_4:def 3;
            y in C by A25,Th15;
            then [y,x] in R or x = y or [x,y] in R by A16,Def3;
            hence y <= x by A18,A23,A26,WAYBEL_4:def 3,def 4;
          end;
          then a <= x by A3,A5,YELLOW_0:def 9;
          hence thesis by A8,A24,WAYBEL_4:def 4;
        end;
      end;
    end;
  end;
end;
