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 NTC:
  for a,b be non trivial Nat holds
    a,b are_coprime implies not a divides b & not b divides a
  proof
    let a,b be non trivial Nat;
    assume a,b are_coprime; then
    1 = a gcd (b mod a) & 1 = b gcd (a mod b) by NAT_D:28; then
    A3: not (b mod a) = 0 & not (a mod b) = 0 by Def0;
    a = (a div b)*b + (a mod b) & b = (b div a)*a + (b mod a) by NAT_D:2; then
    not a = (a div b)*b & not b = (b div a)*a by A3;
    hence thesis by NAT_D:3;
  end;
