reserve x for object,
  a,b for Real,
  k,k1,i1,j1,w for Nat,
  m,m1,n,n1 for Integer;
reserve p,q for Rational;

theorem Th5:
  ex m,k st k > 0 & p = m/k
proof
  consider m,n such that
A1: n > 0 and
A2: p = m/n by Th2;
  per cases by A1;
  suppose
    0<n;
    then n is Element of NAT by INT_1:3;
    hence thesis by A1,A2;
  end;
  suppose
A3: n<0;
A4: p = -(-m)/n by A2
      .= (-m)/(-n) by XCMPLX_1:188;
A5: -n<>0 by A1;
    -n is Element of NAT by A3,INT_1:3;
    hence thesis by A4,A5;
  end;
end;
