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
  pfexp (n|^a) = a * pfexp n
proof
  for i being object st i in SetPrimes
    holds (pfexp (n|^a)).i = (a * pfexp n) .i
  proof
    let i be object;
    assume i in SetPrimes;
    then reconsider x = i as prime Element of NAT by NEWTON:def 6;
    thus (pfexp (n|^a)).i = x |-count (n|^a) by Def8
      .= a * (x |-count n) by Th32
      .= a * (pfexp n).i by Def8
      .= (a * pfexp n).i by Def2;
  end;
  hence thesis;
end;
