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 DIC:
  for a be non trivial Nat, b,c be non zero Integer holds
  a |-count b = a |-count c iff
  (for n be Nat holds a|^n divides b iff a|^n divides c)
  proof
    let a be non trivial Nat, b,c be non zero Integer;
    A1: a > 1 by Def0;
    a |-count b <> a |-count c implies (ex n be Nat st
    (a|^n divides b & not a|^n divides c) or
      (a|^n divides c & not a|^n divides b))
    proof
      reconsider n = a |-count b, m = a |-count c as Nat;
      assume a|-count b <> a |-count c; then
      per cases by XXREAL_0:1;
      suppose n > m; then
        a|^n divides b & not a|^n divides c by DL,A1,Def6;
        hence thesis;
      end;
      suppose n < m; then
        m >= n+1 by NAT_1:13; then
        a|^(n+1) divides a|^m & a|^m divides c by NEWTON:89,Def6,A1; then
        a|^(n+1) divides c & not a|^(n+1) divides b by INT_2:9,A1,Def6;
        hence thesis;
      end;
    end;
    hence thesis by DN;
  end;
