reserve A,B,C for Ordinal;
reserve a,b,c,d for natural Ordinal;
reserve l,m,n for natural Ordinal;
reserve i,j,k for Element of omega;
reserve x,y,z for Element of RAT+;

theorem Th34:
  numerator x, denominator x are_coprime
proof
  per cases;
  suppose
    x in omega;
    then denominator x = 1 by Def9;
    hence thesis by Th2;
  end;
  suppose
A1: not x in omega;
    then consider i,j such that
A2: x = [i,j] and
A3: i,j are_coprime and
    j <> {} and
    j <> 1 by Th29;
    i = numerator x by A1,A2,Def8;
    hence thesis by A1,A2,A3,Def9;
  end;
end;
