reserve m, n for Nat;

theorem
  for n being non zero Nat holds Radical n = 1 iff n = 1
proof
  let n be non zero Nat;
  thus Radical n = 1 implies n = 1
  proof
A1: rng PFactors n c= NAT by VALUED_0:def 6;
    consider f being FinSequence of COMPLEX such that
A2: Product PFactors n = Product f and
A3: f = (PFactors n) * canFS (support PFactors n) by NAT_3:def 5;
    rng f c= rng PFactors n by A3,RELAT_1:26;
    then rng f c= NAT by A1;
    then reconsider f as FinSequence of NAT by FINSEQ_1:def 4;
    assume
A4: Radical n = 1;
    assume n <> 1;
    then consider p being Prime such that
A5: p divides n by Th5;
A6: p in support pfexp n by A5,NAT_3:37;
    then
A7: p in support PFactors n by Def6;
    then p in rng canFS support PFactors n by FUNCT_2:def 3;
    then consider y being object such that
A8: y in dom canFS support PFactors n & p = (canFS support PFactors n
    ).y by FUNCT_1:def 3;
    (PFactors n).p = p by A6,Def6;
    then
A9: f.y = p by A3,A8,FUNCT_1:13;
    support PFactors n c= dom PFactors n by PRE_POLY:37;
    then y in dom f by A3,A7,A8,FUNCT_1:11;
    then 1 < p & p in rng f by A9,FUNCT_1:3,INT_2:def 4;
    hence thesis by A4,A2,NAT_3:7,NAT_D:7;
  end;
  thus thesis by Th45,NAT_3:20;
end;
