reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r,s for Real;
reserve p,p1,p2,p3 for Prime;

theorem Th33:
  for n being non zero Nat holds
  Product (EmptyBag SetPrimes +* Euler_factorization_1 n) divides n
  proof
    let n be non zero Nat;
    set E = Euler n;
    set b0 = ppf n;
    set F1 = Euler_factorization_1 n;
    set b1 = B+*F1;
    consider f0 being FinSequence of COMPLEX such that
A1: Product b0 = Product f0 and
A2: f0 = b0*canFS(support b0) by NAT_3:def 5;
    consider f1 being FinSequence of COMPLEX such that
A3: Product b1 = Product f1 and
A4: f1 = b1*canFS(support b1) by NAT_3:def 5;
A5: support b0 = support pfexp n by NAT_3:def 9;
A6: support b1 = support F1 by Th12;
A7: support F1 = dom F1 by Th28;
A8: dom F1 = support b0 by Def2;
A9: dom f0 = dom canFS(support b0) by A2,Th13;
A10: rng canFS(support b1) c= support b1 by FINSEQ_1:def 4;
A11: dom f1 = dom canFS(support b1) by A4,Th13;
     then
A12: len f0 = len f1 by A6,A7,A8,A9,FINSEQ_3:29;
    rng f0 c= NAT by A2,ORDINAL1:def 12;
    then reconsider f0 as FinSequence of NAT by FINSEQ_1:def 4;
    rng f1 c= NAT by A4,ORDINAL1:def 12;
    then reconsider f1 as FinSequence of NAT by FINSEQ_1:def 4;
    for x being Nat st 1 <= x <= len f1 holds f1.x divides f0.x
    proof
      let x be Nat such that
A13:  1 <= x and
A14:  x <= len f1;
      set p = (canFS(support b1)).x;
A15:  x in dom f1 by A13,A14,FINSEQ_3:25;
      dom(b1*canFS(support b1)) c= dom canFS(support b1) by RELAT_1:25;
      then
A16:  p in rng canFS(support b1) by A4,A15,FUNCT_1:def 3;
      then consider c being non zero Nat such that
A17:  c = p |-count n and
A18:  F1.p = p|^(c-1) by A6,A7,A10,Def2;
A19:  f1.x = b1.p by A4,A15,FUNCT_1:12
      .= p|^(c-1) by A6,A7,A10,A16,A18,FUNCT_4:13;
A20:  f0.x = b0.p by A2,A6,A7,A8,A9,A11,A13,A14,FINSEQ_3:25,FUNCT_1:12
      .= p|^c by A5,A6,A7,A8,A10,A16,A17,NAT_3:def 9;
      c-1 <= c-0 by XREAL_1:7;
      hence f1.x divides f0.x by A19,A20,NEWTON:89;
    end;
    then Product f1 divides Product f0 by A12,Th32;
    hence thesis by A1,A3,NAT_3:61;
  end;
