
theorem Th12:
  for p be G_RAT ex x, y be G_INTEG st y <> 0 & p = x/y
  proof
    let p be G_RAT;
    reconsider Rp = Re p, Ip = Im p as Rational;
    consider m1, n1 be Integer such that
    A1: n1 > 0 & Rp = m1/n1 by RAT_1:2;
    consider m2, n2 be Integer such that
    A2: n2 > 0 & Ip = m2/n2 by RAT_1:2;
    set z = n1*n2;
    Re z = z & Im z = 0 by COMPLEX1:def 1,COMPLEX1:def 2;
    then A3: (Re(z * p) = z * Rp - 0 * Ip) & (Im(z * p) = z * Ip + Rp * 0)
    by COMPLEX1:9;
    A4: Re(z*p) = n1/n1*m1*n2 by A1,A3
    .= 1*m1*n2 by XCMPLX_1:60,A1
    .= m1*n2;
    A5: Im(z*p) = n2/n2*m2*n1 by A2,A3
    .= 1*m2*n1 by XCMPLX_1:60,A2
    .= m2*n1;
    reconsider x = z*p as G_INTEG by A4,A5,Lm1;
    take x,z;
    x/z = z/z * p
    .= 1*p by XCMPLX_1:60,A1,A2
    .= p;
    hence thesis by A1,A2;
  end;
