reserve x,y for object, X for set;

theorem Th14:
  for p be bag of SetPrimes st p is prime-factorization-like holds
  Product p = 1 iff support p = {}
proof
  let p be bag of SetPrimes;
  assume
A1: p is prime-factorization-like;
A2: now
    assume
A3: Product p = 1;
    thus support p = {}
    proof
      assume support p <> {};
      then consider q be object such that
A4:   q in support p by XBOOLE_0:def 1;
      q in SetPrimes by A4;
      then reconsider q as Element of NAT;
      reconsider q as Prime by A4,NEWTON:def 6;
      q =1 by A1,A3,A4,Lm4,WSIERP_1:15;
      hence contradiction by INT_2:def 4;
    end;
  end;
  now
    consider f being FinSequence of COMPLEX such that
A5: Product p = Product f and
A6: f = p*canFS(support p) by NAT_3:def 5;
    assume support p = {};
    hence Product p =1 by A5,A6,RVSUM_1:94;
  end;
  hence thesis by A2;
end;
