
theorem Th19:
  for i,j,k being Integer st i,j are_coprime holds i gcd (j
  * k) = i gcd k
proof
  let i,j,k be Integer;
  assume
A1: i gcd j = 1;
  (i*k) gcd (j*k) = |.k.| * (i gcd j) by Th16;
  then i gcd |.k.| = (i gcd (i*k)) gcd (j*k) by A1,Th18
    .= |.i.| gcd (j*k) by Th17
    .= (j*k) gcd i by Th14;
  hence thesis by Th14;
end;
