reserve a, b, n for Nat,
  r for Real,
  f for FinSequence of REAL;
reserve p for Prime;

theorem Th32:
  for a being non zero Nat holds p |-count (a|^b) = b * (p |-count a)
proof
  let a be non zero Nat;
A1: p <> 1 by INT_2:def 4;
  defpred P[Nat] means p |-count (a|^$1) = $1 * (p |-count a);
A2: for x being Nat st P[x] holds P[x+1]
  proof
    let x be Nat such that
A3: P[x];
    thus p |-count (a|^(x+1)) = p |-count (a|^x*a) by NEWTON:6
      .= x * (p |-count a) + 1 * (p |-count a) by A3,Th28
      .= (x+1) * (p |-count a);
  end;
  p |-count (a|^0) = p |-count 1 by NEWTON:4
    .= 0 * (p |-count a) by A1,Th21;
  then
A4: P[ 0 ];
  for x being Nat holds P[x] from NAT_1:sch 2(A4,A2);
  hence thesis;
end;
