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 DL:
  for a be non trivial Nat, b,c be non zero Integer holds
  a |-count b > a |-count c implies a|^(a |-count c) divides b &
  not a|^(a |-count b) divides c
  proof
    let a be non trivial Nat, b,c be non zero Integer;
A1: a > 1 by Def0;
    reconsider n = a |-count b, m = a |-count c as Nat;
    assume
A2: a |-count b > a |-count c; then
    n >= m+1 by NAT_1:13; then
    consider k such that
A3: n = (m+1)+k by NAT_1:10;
    not a|^(m+1) divides c & a|^(m+1) divides a|^(m+1)*a|^k by A1,Def6; then
    not a|^(m+1)*a|^k divides c by INT_2:9; then
    not a|^n divides c & a|^n divides b & a|^m divides a|^n
      by NEWTON:8,A3,A1,Def6,A2,NEWTON:89;
    hence thesis by INT_2:9;
  end;
