reserve x,y for object, X for set;

theorem Th13:
  for p be bag of SetPrimes st p is prime-factorization-like holds
  Product p <> 0
proof
  let p be bag of SetPrimes;
  assume
A1: p is prime-factorization-like;
A2: rng canFS(support p) = support p by FUNCT_2:def 3;
  consider f being FinSequence of COMPLEX such that
A3: Product p = Product f and
A4: f = p*canFS(support p) by NAT_3:def 5;
  reconsider f as complex-valued FinSequence;
  assume Product p = 0;
  then consider k be Nat such that
A5: k in dom f and
A6: f.k = 0 by A3,RVSUM_1:103;
  k in dom (canFS(support p)) by A4,A5,FUNCT_1:11;
  then
A7: (canFS(support p)).k in support p by A2,FUNCT_1:3;
  then reconsider q= (canFS(support p)).k as Prime by NEWTON:def 6;
  ex n be Nat st 0 < n & p.q = q|^n by A1,A7;
  hence contradiction by A4,A5,A6,FUNCT_1:12;
end;
