reserve x,y for object, X for set;

theorem Th12:
  for p,q be bag of SetPrimes st p is prime-factorization-like & q
  is prime-factorization-like & (support p) misses (support q) holds Product p,
  Product q are_coprime
proof
  let p,q be bag of SetPrimes;
  assume that
A1: p is prime-factorization-like and
A2: q is prime-factorization-like and
A3: (support p) misses (support q);
  assume not Product p,Product q are_coprime;
  then consider x be Prime such that
A4: x divides (Product p) and
A5: x divides (Product q) by PYTHTRIP:def 2;
A6: x in (support q) by A2,A5,Lm7;
  x in (support p) by A1,A4,Lm7;
  hence contradiction by A3,A6,XBOOLE_0:3;
end;
