
theorem Th09:
  for X1,X2 be set, S1 be semialgebra_of_sets of X1,
      S2 be semialgebra_of_sets of X2 holds
  the set of all [:A,B:] where A is Element of S1, B is Element of S2
    is semialgebra_of_sets of [:X1,X2:]
proof
   let X1,X2 be set,
   S1 be semialgebra_of_sets of X1, S2 be semialgebra_of_sets of X2;
   set S = the set of all [:A,B:] where A is Element of S1, B is Element of S2;
A1:S is Semiring of [:X1,X2:] by Th08;
   X1 in S1 & X2 in S2 by SRINGS_3:def 6; then
   [:X1,X2:] in S;
   hence thesis by A1,SRINGS_3:def 6;
end;
