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 CD:
  for b be non zero Nat, a be non trivial Nat holds
    a divides b iff a |-count (a gcd b) = 1
  proof
    let b be non zero Nat, a be non trivial Nat;
    reconsider c = a gcd b as non zero Nat;
    A1: a > 1 by Def0;
    A2: c divides b by INT_2:def 2;
    thus a divides b implies a |-count (a gcd b) = 1
    proof
      assume a divides b; then
      a gcd b = |.a.| by NEWTON02:3;
      hence thesis by NAT_3:22,Def0;
    end;
    assume a|-count (a gcd b) = 1; then
    a|^1 divides c by A1,NAT_3:def 7;
    hence thesis by A2,INT_2:9;
  end;
