reserve m, n for Nat;

theorem Th52:
  for n being non zero Nat holds Radical n > 0
proof
  let n be non zero Nat;
  assume Radical n <= 0;
  then Product PFactors n = 0;
  then consider f being FinSequence of COMPLEX such that
A1: Product f = 0 and
A2: f = (PFactors n) * canFS(support PFactors n) by NAT_3:def 5;
  not ex k being Nat st k in dom f & f.k = 0
  proof
    given k being Nat such that
A3: k in dom f and
A4: f.k = 0;
    k in dom canFS support PFactors n by A2,A3,FUNCT_1:11;
    then
A5: rng canFS support PFactors n c= support PFactors n & (canFS support
    PFactors n).k in rng canFS support PFactors n by FINSEQ_1:def 4,FUNCT_1:3;
    then (canFS support PFactors n).k in support PFactors n;
    then reconsider
    p = (canFS support PFactors n).k as prime Element of NAT by NEWTON:def 6;
    p in support PFactors n by A5;
    then
A6: p in support pfexp n by Def6;
    f.k = (PFactors n).p by A2,A3,FUNCT_1:12
      .= p by A6,Def6;
    hence thesis by A4;
  end;
  hence contradiction by A1,RVSUM_1:103;
end;
