reserve x,y for object, X for set;

theorem Th7:
  for p,q be bag of SetPrimes st (support p) c= (support q) & p | (
  support p) = q | (support p) holds (Product p) divides (Product q)
proof
  let p,q be bag of SetPrimes;
  assume that
A1: (support p) c= (support q) and
A2: p | (support p) =q | (support p);
  consider r be bag of SetPrimes such that
  support r =(support q) \ (support p) and
A3: (support p) misses (support r) and
  r | (support r) = q| ( support r) and
A4: p+r = q by A1,A2,Lm2;
  (Product p)*(Product r) = Product (q) by A3,A4,NAT_3:19;
  hence thesis by INT_1:def 3;
end;
