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
  for a be non trivial Nat, n,b be non zero Nat holds
    a |-count b >= n*(a|^n |-count b)
  proof
    let a be non trivial Nat, n,b be non zero Nat;
A1: a <> 1 & b <> 0 by Def0;
    reconsider k = a|^n as non trivial Nat;
    k <> 1 & b <> 0 by Def0; then
    k|^(k |-count b) divides b & not k|^((k |-count b)+1) divides b
      by NAT_3:def 7; then
    a|^(n*(a|^n |-count b)) divides b by NEWTON:9;
    hence thesis by A1,NAT330;
  end;
