
theorem
  for n be non zero Nat, Xn be non-empty n-element FinSequence,
      X1 be non-empty 1-element FinSequence,
      Sn be SemialgebraFamily of Xn, S1 be SemialgebraFamily of X1 holds
   SemiringProduct(Sn^S1) is semialgebra_of_sets of product(Xn^X1)
proof
  let n be non zero Nat, Xn be non-empty n-element FinSequence,
       X1 be non-empty 1-element FinSequence,
       Sn be SemialgebraFamily of Xn, S1 be SemialgebraFamily of X1;
A1:SemiringProduct(Sn) is semialgebra_of_sets of product Xn &
   SemiringProduct(S1) is semialgebra_of_sets of product X1 by Th15;
   reconsider S12 = the set of all [:s1,s2:] where
   s1 is Element of SemiringProduct(Sn),
   s2 is Element of SemiringProduct(S1) as
   semialgebra_of_sets of [:product Xn,product X1:] by A1,Th14;
   SemiringProduct(Sn) is cap-closed semiring_of_sets of product Xn &
   SemiringProduct(S1) is cap-closed semiring_of_sets of product X1
     by SRINGS_4:38; then
A5:SemiringProduct(Sn^S1) is cap-closed semiring_of_sets of product (Xn^X1)
     by SRINGS_4:37; then
A6:SemiringProduct(Sn^S1) is semi-diff-closed by SRINGS_3:10;
A11:dom Xn = dom Sn & dom X1 = dom S1 by SRINGS_4:18; then
A8:len Xn = len Sn & len X1 = len S1 by FINSEQ_3:29;
   len(Xn^X1) = len Xn + len X1 & len(Sn^S1) = len Sn + len S1
     by FINSEQ_1:22; then
A7:dom (Xn^X1) = dom (Sn^S1) by A8,FINSEQ_3:29;
   now let x be object;
    assume A9: x in dom(Sn^S1);
    per cases by A9,FINSEQ_1:25;
    suppose A10: x in dom Sn; then
     x in Seg n by FINSEQ_1:89; then
     Xn.x in Sn.x by Th11; then
     (Xn^X1).x in Sn.x by A10,A11,FINSEQ_1:def 7;
     hence (Xn^X1).x in (Sn^S1).x by A10,FINSEQ_1:def 7;
    end;
    suppose ex k be Nat st k in dom S1 & x = len Sn + k; then
     consider k be Nat such that
A12:  k in dom S1 & x = len Sn + k;
     k in Seg 1 by A12,FINSEQ_1:89; then
     X1.k in S1.k by Th11; then
     (Xn^X1).x in S1.k by A12,A11,A8,FINSEQ_1:def 7;
     hence (Xn^X1).x in (Sn^S1).x by A12,FINSEQ_1:def 7;
    end;
   end; then
   Xn^X1 in product(Sn^S1) by A7,CARD_3:9; then
   product(Xn^X1) in SemiringProduct(Sn^S1) by SRINGS_4:def 4;
   hence thesis by A5,A6,SRINGS_3:def 6;
  end;
