
theorem
  RAT+ c= RAT
  proof
    not [0,0] in RAT+
    proof
      assume
A1:   [0,0] in RAT+;
      reconsider i = 0 as Element of omega by ORDINAL1:def 12;
      i,i are_coprime & i <> {} & i <> 1 by A1,ARYTM_3:33;
      hence contradiction;
    end;
    hence thesis by XBOOLE_1:7,ZFMISC_1:34,NUMBERS:def 3;
  end;
