reserve m, n for Nat;

theorem Th54:
  for p being Prime, n being non zero Nat holds Radical (p |^ n) = p
proof
  let p be Prime, n be non zero Nat;
  reconsider p as prime Element of NAT by ORDINAL1:def 12;
  reconsider f = <*p*> as FinSequence of NAT;
  set s = p |^ n;
A1: f = (PFactors s) * <*p*> by Th47
    .= (PFactors s) * canFS({p}) by FINSEQ_1:94;
  rng f c= NAT by FINSEQ_1:def 4;
  then rng f c= COMPLEX by NUMBERS:20;
  then
A2: Product f = p & f is FinSequence of COMPLEX by FINSEQ_1:def 4,RVSUM_1:95;
  support PFactors s = support pfexp s by Def6
    .= support pfexp p by NAT_3:48
    .= {p} by NAT_3:43;
  hence thesis by A1,A2,NAT_3:def 5;
end;
