reserve X1,X2,X3,X4 for set;
reserve n for non zero Nat;
reserve X for non-empty n-element FinSequence;

theorem Thm35:
  for X1,X2 being set, S1 being cap-closed semiring_of_sets of X1,
  S2 being cap-closed semiring_of_sets of X2 holds
  the set of all [:s1,s2:] where s1 is Element of S1,
  s2 is Element of S2 is cap-closed semiring_of_sets of [:X1,X2:]
  proof
    let X1,X2 be set, S1 be cap-closed semiring_of_sets of X1,
    S2 be cap-closed semiring_of_sets of X2;
    set S = {s where s is Subset of [:X1,X2:]:ex x1,x2 be set st x1 in S1 &
    x2 in S2 & s=[:x1,x2:]};
    S is semiring_of_sets of [:X1,X2:] & S is cap-closed
    by SRINGS_2:7,SRINGS_2:3;
    hence thesis by SRINGS_2:2;
  end;
