theorem Th50:
  for T being Tree, C being Chain of T, t being Element of T st t
  in C & not C is finite ex t9 being Element of T st t9 in C & t
  is_a_proper_prefix_of t9
proof
  let T be Tree, C be Chain of T, t be Element of T such that
A1: t in C and
A2: not C is finite;
  now
    assume
A3: not ex t9 being Element of T st t9 in C & t is_a_proper_prefix_of t9;
A4: for t9 being Element of T st t9 in C holds t9 is_a_prefix_of t
    proof
      let t9 be Element of T such that
A5:   t9 in C;
      now
        assume
A6:     not t9 is_a_prefix_of t;
        t,t9 are_c=-comparable by A1,A5,TREES_2:def 3;
        then t is_a_prefix_of t9 by A6;
        then t is_a_proper_prefix_of t9 by A6;
        hence contradiction by A3,A5;
      end;
      hence thesis;
    end;
    C c= ProperPrefixes t \/ {t}
    proof
      let x be object;
      assume
A7:   x in C;
      then reconsider t9 = x as Element of T;
A8:   t9 is_a_prefix_of t by A4,A7;
      t9 in ProperPrefixes t \/ {t}
      proof
        per cases by A8;
        suppose
          t9 is_a_proper_prefix_of t;
          then t9 in ProperPrefixes t by TREES_1:12;
          hence thesis by XBOOLE_0:def 3;
        end;
        suppose
          t9 = t;
          then t9 in {t} by TARSKI:def 1;
          hence thesis by XBOOLE_0:def 3;
        end;
      end;
      hence thesis;
    end;
    hence contradiction by A2;
  end;
  hence thesis;
end;
