
theorem Th08:
  for X1,X2 be set, S1 be Semiring of X1, S2 be Semiring of X2 holds
    the set of all [:A,B:] where A is Element of S1, B is Element of S2
      is Semiring of [:X1,X2:]
proof
   let X1,X2 be set, S1 be Semiring of X1, S2 be Semiring of X2;
A1:S1 is semiring_of_sets of X1 by SRINGS_3:9;
   S2 is diff-c=-finite-partition-closed by SRINGS_3:9; then
   the set of all [:A,B:] where A is Element of S1, B is Element of S2
    is cap-closed semiring_of_sets of [:X1,X2:] by A1,SRINGS_4:36;
   hence thesis by SRINGS_3:10;
end;
