reserve a,b,c,d,m,x,n,j,k,l for Nat,
  t,u,v,z for Integer,
  f,F for FinSequence of NAT;
reserve p,q,r,s for real number;

theorem
  k>0 implies a gcd b <= a gcd b*k
  proof
    assume
    k>0; then
    b*k = 0 iff b = 0; then
    A1: a gcd b*k = 0 implies a gcd b = 0 by INT_2:5;
    A2: a gcd b divides b & a gcd b divides a by INT_2:def 2; then
    a gcd b divides b*k by INT_2:2;
    hence thesis by A1,A2,INT_2:22,27;
  end;
