reserve m, n for Nat;

theorem Th56:
  for p being Prime, n being non zero Nat holds p
  divides n iff p divides Radical n
proof
  let p be Prime, n be non zero Nat;
  thus p divides n implies p divides Radical n
  proof
    assume that
A1: p divides n and
A2: not p divides Radical n;
A3: p in support pfexp n by A1,NAT_3:37;
    then
A4: 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
A5: y in dom canFS support PFactors n and
A6: p = (canFS support PFactors n).y by FUNCT_1:def 3;
    consider f being FinSequence of COMPLEX such that
A7: Product PFactors n = Product f and
A8: f = (PFactors n) * canFS support PFactors n by NAT_3:def 5;
    rng PFactors n c= NAT & rng f c= rng PFactors n by A8,RELAT_1:26
,VALUED_0:def 6;
    then
A9: rng f c= NAT;
    support PFactors n c= dom PFactors n by PRE_POLY:37;
    then
A10: y in dom f by A4,A8,A5,A6,FUNCT_1:11;
    reconsider f as FinSequence of NAT by A9,FINSEQ_1:def 4;
    (PFactors n).p = p by A3,Def6;
    then f.y = p by A8,A6,A10,FUNCT_1:12;
    then p in rng f by A10,FUNCT_1:3;
    hence contradiction by A2,A7,NAT_3:7;
  end;
  assume
A11: p divides Radical n;
  Radical n divides n by Th55;
  hence thesis by A11,NAT_D:4;
end;
