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
  m1 = denominator p & m2 = denominator q &
  n1 = numerator p & n2 = numerator q & n2 <> 0 implies
  denominator(p/q) = (m1*n2) / ( (n1*m2) gcd (m1*n2) ) &
  numerator(p/q) = (n1*m2) / ( (n1*m2) gcd (m1*n2) )
  proof
    assume
A1: m1 = denominator p & m2 = denominator q &
    n1 = numerator p & n2 = numerator q & n2 <> 0;
    hence denominator(p/q) = (m1*n2) div ( (n1*m2) gcd (m1*n2) ) by Th41
    .= (m1*n2) / ( (n1*m2) gcd (m1*n2) ) by Th8;
    thus numerator(p/q) = (n1*m2) div ( (n1*m2) gcd (m1*n2) ) by A1,Th41
    .= (n1*m2) / ( (n1*m2) gcd (m1*n2) ) by Th7;
  end;
