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
  m = denominator p & n = denominator q &
  i = numerator p & j = numerator q implies
  denominator(p*q) = (m*n) / ( (i*j) gcd (m*n) ) &
  numerator(p*q) = (i*j) / ( (i*j) gcd (m*n) )
  proof
    assume
A1: m = denominator p & n = denominator q &
    i = numerator p & j = numerator q;
    hence denominator(p*q) = (m*n) div ( (i*j) gcd (m*n) ) by Th39
    .= (m*n) / ( (i*j) gcd (m*n) ) by Th8;
    thus numerator(p*q) = (i*j) div ( (i*j) gcd (m*n) ) by A1,Th39
    .= (i*j) / ( (i*j) gcd (m*n) ) by Th7;
  end;
