reserve a,b,c,d,x,j,k,l,m,n,o,xi,xj for Nat,
  p,q,t,z,u,v for Integer,
  a1,b1,c1,d1 for Complex;

theorem
  for a,b,c be Integer st a,b are_coprime holds
    c gcd (a*b) = (c gcd a)*(c gcd b)
  proof
    let a,b,c be Integer such that
    A1: a,b are_coprime;
    reconsider k = |.a.|, l = |.b.|, m = |.c.| as Nat;
    A2: k,l are_coprime by A1,INT_2:34;
    |.a.|*|.b.| = |.a*b.| by COMPLEX1:65; then
    c gcd (a*b) = m gcd (k*l) & c gcd a = m gcd k & c gcd b = m gcd l
      by INT_2:34;
    hence thesis by A2,NAT517;
  end;
