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, b be non zero Integer holds
    a |-count (a|^n*b) = n + a|-count b
  proof
    let a be non trivial Nat, b be non zero Integer;
A0: a|^n*a|^(a |-count b) = a|^((a |-count b)+n) &
      a|^n*a|^((a|-count b)+1) = a|^(((a |-count b)+1)+n) by NEWTON:8;
A1: a <> 1 by Def0; then
    a|^(a |-count b) divides b &
      not a|^((a|-count b)+1) divides b by Def6; then
    a|^((a |-count b)+n) divides (a|^n*b) &
      not a|^(((a |-count b)+n)+1) divides (a|^n*b) by INT_4:7,A0;
    hence thesis by A1,Def6;
  end;
