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 Count2:
  for a be non trivial Nat, b be non zero Integer, n be non zero Nat holds
    n*(a |-count b) <= a |-count b|^n < n*((a |-count b) + 1)
  proof
    let a be non trivial Nat, b be non zero Integer, n be non zero Nat;
    reconsider k = a |-count b as Nat;
    A0: a|^k|^n = a|^(k*n) & a|^(k+1)|^n = a|^((k+1)*n) by NEWTON:9;
    a <> 1 by NAT_2:def 1; then
    A1: a|^k divides b & not a|^(k+1) divides b by Def6;
    a|^k gcd b = |.a|^k.| & a|^(k+1) gcd b <> |.a|^(k+1).|
      by A1,NEWTON02:3; then
    (a|^k gcd b)|^n = (a|^k)|^n & (a|^(k+1) gcd b)|^n <> (a|^(k+1))|^n
      by WSIERP_1:3; then
    (a|^k|^n) gcd (b|^n) = |.a|^k|^n.| &
      (a|^(k+1)|^n) gcd (b|^n) <> |.a|^(k+1)|^n.| by NEWTON027;
    hence thesis by Count1,A0,NEWTON02:3;
  end;
