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 p be prime Nat, a be non trivial Nat, b be non zero Nat holds
    a |-count (p|^b) <= b
proof
  let p be prime Nat, a be non trivial Nat, b be non zero Nat;
  A1: a > 1 by Def0;
  per cases;
  suppose p = a;
    hence thesis by Def0,NAT_3:25;
  end;
  suppose
    p <> a; then
    a |-count p = 0 by A1,NAT_3:24; then
    not a|^(0+b) divides p|^b by MOB16,WSIERP_1:26;
    hence thesis by Count1;
  end;
end;
