
theorem Th5:
  for n being Ordinal, T being connected TermOrder of n, b1,b2
  being bag of n holds b1 <= b2,T iff not b2 < b1,T
proof
  let n be Ordinal, T be connected TermOrder of n, b1,b2 be bag of n;
A1: T is_connected_in field T by RELAT_2:def 14;
  per cases;
  suppose
    b1 = b2;
    hence thesis by Lm2;
  end;
  suppose
A2: b1 <> b2;
A3: not b2 < b1,T implies b1 <= b2,T
    proof
      assume
A4:   not b2 < b1,T;
      now
        per cases by A4;
        case
A5:       not b2 <= b1,T;
A6:       b1 in field T & b2 in field T by Lm4;
          not [b2,b1] in T by A5;
          then [b1,b2] in T by A1,A2,A6,RELAT_2:def 6;
          hence thesis;
        end;
        case
          b1 = b2;
          hence thesis by A2;
        end;
      end;
      hence thesis;
    end;
    b1 <= b2,T implies not b2 < b1,T
    by Lm3;
    hence thesis by A3;
  end;
end;
