
theorem Th14:
  for X1,X2 be set, S1 be semialgebra_of_sets of X1,
      S2 be semialgebra_of_sets of X2 holds
   the set of all [:s1,s2:] where s1 is Element of S1,
     s2 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 [:s1,s2:] where s1 is Element of S1,
           s2 is Element of S2;
   S1 is cap-closed semiring_of_sets of X1 &
   S2 is cap-closed semiring_of_sets of X2 by SRINGS_3:9; then
   S is cap-closed semiring_of_sets of [:X1,X2:] by SRINGS_4:36; then
A1:S is with_empty_element semi-diff-closed cap-closed
        Subset-Family of [:X1,X2:] by SRINGS_3:10;
   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;
