reserve p,q for Rational;
reserve g,m,m1,m2,n,n1,n2 for Nat;
reserve i,i1,i2,j,j1,j2 for Integer;

theorem Th23:
  m <> 0 & n <> 0 implies
  denominator(i/m-j/n) = (m*n) div ( (i*n-j*m) gcd (m*n) ) &
  numerator(i/m-j/n) = (i*n-j*m) div ( (i*n-j*m) gcd (m*n) )
  proof
    assume that
A1: m <> 0 and
A2: n <> 0;
    (i/m-j/n) = (i*n)/(m*n) - j/n by A2,XCMPLX_1:91
    .= (i*n)/(m*n) - (j*m)/(m*n) by A1,XCMPLX_1:91
    .= (i*n-j*m)/(m*n);
    hence thesis by A1,A2,Th15;
  end;
