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

theorem
  ppf (n|^m) = (ppf n) |^ m
proof
  now
    let i be object;
A1: m >= 0+1 by NAT_1:13;
A2: ((ppf n) |^ m).i = (ppf n).i |^ m by Def6;
    assume i in SetPrimes;
    then reconsider p = i as prime Element of NAT by NEWTON:def 6;
A3: p |-count (n |^ m) = m * (p |-count n) by Th32;
    per cases;
    suppose
A4:   p |-count n = 0;
      hence (ppf (n|^m)).i = 0 by A3,Th55
        .= (0 qua Nat) |^ m by A1,NEWTON:11
        .= ((ppf n) |^ m).i by A2,A4,Th55;
    end;
    suppose
A5:   p |-count n <> 0;
      hence (ppf (n|^m)).i = p |^ (p |-count (n |^ m)) by A3,Th56
        .= (p |^ (p |-count n)) |^ m by A3,NEWTON:9
        .= ((ppf n) |^ m).i by A2,A5,Th56;
    end;
  end;
  hence thesis;
end;
