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 Th37:
  x <> {} iff numerator x <> {}
proof
  hereby
    assume that
A1: x <> {} and
A2: numerator x = {};
A3: not x in omega by A1,A2,Def8;
    then consider i,j such that
A4: x = [i,j] and
A5: i,j are_coprime and
    j <> {} and
A6: j <> 1 by Th29;
    i = {} by A2,A3,A4,Def8;
    hence contradiction by A5,A6,Th3;
  end;
  {} in omega by ORDINAL1:def 11;
  hence thesis by Def8;
end;
