reserve m, n for Nat;

theorem Th45:
  PFactors 1 = EmptyBag SetPrimes
proof
  set f = PFactors 1;
  for z being object st z in dom f holds f.z = 0
  proof
    let z be object;
    assume z in dom f;
    then z in SetPrimes by PARTFUN1:def 2;
    then reconsider z as Element of NAT;
    support pfexp 1 = {};
    then not z in support f by Def6;
    hence thesis by PRE_POLY:def 7;
  end;
  then
A1: f = (dom f) --> 0 by FUNCOP_1:11;
  dom f = SetPrimes by PARTFUN1:def 2;
  hence thesis by A1,PBOOLE:def 3;
end;
