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
  m divides n implies pfexp (n div m) = pfexp n -' pfexp m
proof
  assume
A1: m divides n;
  for i being object st i in SetPrimes holds (pfexp (n div m)).i = (pfexp n
  -' pfexp m).i
  proof
    let i be object;
    assume i in SetPrimes;
    then reconsider a = i as prime Element of NAT by NEWTON:def 6;
    thus (pfexp (n div m)).i = a |-count (n div m) by Def8
      .= (a |-count n) -' (a |-count m) by A1,Th31
      .= (pfexp n).i -' (a |-count m) by Def8
      .= (pfexp n).i -' (pfexp m).i by Def8
      .= (pfexp n -' pfexp m).i by PRE_POLY:def 6;
  end;
  hence thesis;
end;
