
theorem Th15:
  for n be non zero Nat, X be non-empty n-element FinSequence,
      S be SemialgebraFamily of X holds
    SemiringProduct(S) is semialgebra_of_sets of product X
proof
   let n be non zero Nat, X be non-empty n-element FinSequence,
       S be SemialgebraFamily of X;
   defpred P[non zero Nat] means
    for X be non-empty $1-element FinSequence,
    S be SemialgebraFamily of X holds
    SemiringProduct(S) is semialgebra_of_sets of product X;
A1:P[1] by Th13;
A2:now let k be non zero Nat;
    assume P[k];
    now let Xn1 be non-empty (k+1)-element FinSequence,
            Sn1 be SemialgebraFamily of Xn1;
A3:  SemiringProduct(Sn1) is cap-closed semiring_of_sets of product Xn1
       by SRINGS_4:38; then
A4:  SemiringProduct(Sn1) is semi-diff-closed by SRINGS_3:10;
A6:  dom Xn1 = dom Sn1 by SRINGS_4:18;
     now let x be object;
      assume x in dom Sn1; then
      x in Seg(k+1) by FINSEQ_1:89;
      hence Xn1.x in Sn1.x by Th11;
     end; then
     Xn1 in product Sn1 by A6,CARD_3:9; then
     product Xn1 in SemiringProduct(Sn1) by SRINGS_4:def 4;
     hence SemiringProduct(Sn1) is semialgebra_of_sets of product Xn1
       by A3,A4,SRINGS_3:def 6;
    end;
    hence P[k+1];
   end;
   for k be non zero Nat holds P[k] from NAT_1:sch 10(A1,A2);
   hence thesis;
end;
