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 & i = numerator p implies
  denominator(-p) = m / ( (-i) gcd m ) &
  numerator(-p) = (-i) / ( (-i) gcd m )
  proof
    assume
A1: m = denominator p & i = numerator p;
    hence denominator(-p) = m div ( (-i) gcd m ) by Th31
    .= m / ( (-i) gcd m ) by Th8;
    thus numerator(-p) = (-i) div ( (-i) gcd m ) by A1,Th31
    .= (-i) / ( (-i) gcd m ) by Th7;
  end;
