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 DN:
  for a be non trivial Nat, b,c be non zero Integer holds
  a |-count b = a |-count c & a|^n divides b implies a|^n divides c
  proof
    let a be non trivial Nat, b,c be non zero Integer;
    A0: a > 1 by Def0;
    assume
A1: a |-count b = a |-count c & a|^n divides b;
    not a|^((a|-count b)+1) divides b by A0,Def6; then
    ((a |-count b)+1) > n by A1,PP; then
    a |-count b >= n by NAT_1:13; then
A3: a|^n divides a|^(a |-count b) by NEWTON:89;
    a|^(a |-count c) divides c by A0,Def6;
    hence thesis by A1,A3, INT_2:9;
  end;
