
theorem
  for L being non empty upper-bounded Poset, R being auxiliary(ii) (
Relation of L), C being strict_chain of R, m being Element of L st C is maximal
  & m is_maximum_of C & [m,Top L] in R holds [Top L,Top L] in R & m = Top L
proof
  let L be non empty upper-bounded Poset, R be auxiliary(ii) (Relation of L),
  C be strict_chain of R, m be Element of L such that
A1: C is maximal and
A2: m is_maximum_of C and
A3: [m,Top L] in R;
A4: C c= C \/ {Top L} by XBOOLE_1:7;
  now
A5: m <= Top L by YELLOW_0:45;
    assume
A6: m <> Top L;
A7: {Top L} c= C \/ {Top L} by XBOOLE_1:7;
A8: ex_sup_of C,L by A2;
A9: sup C = m by A2;
    C \/ {Top L} is strict_chain of R
    proof
      let a, b be set;
      assume that
A10:  a in C \/ {Top L} and
A11:  b in C \/ {Top L};
A12:  Top L <= Top L;
      per cases by A10,A11,Lm1;
      suppose
        a in C & b in C;
        hence thesis by Def3;
      end;
      suppose that
A13:    a = Top L and
A14:    b in C;
        reconsider b as Element of L by A14;
        b <= sup C by A8,A14,YELLOW_4:1;
        hence thesis by A3,A9,A12,A13,WAYBEL_4:def 4;
      end;
      suppose that
A15:    a in C and
A16:    b = Top L;
        reconsider a as Element of L by A15;
        a <= sup C by A8,A15,YELLOW_4:1;
        hence thesis by A3,A9,A12,A16,WAYBEL_4:def 4;
      end;
      suppose
        a = Top L & b = Top L;
        hence thesis;
      end;
    end;
    then
A17: C \/ {Top L} = C by A1,A4;
    Top L in {Top L} by TARSKI:def 1;
    then Top L <= sup C by A7,A8,A17,YELLOW_4:1;
    hence contradiction by A6,A5,A9,ORDERS_2:2;
  end;
  hence thesis by A3;
end;
