
theorem Th12:
  for X be non-empty 1-element FinSequence,
      S be SemialgebraFamily of X holds
  the set of all product <*s*> where s is Element of S.1 is
    semialgebra_of_sets of the set of all <*x*> where x is Element of X.1
proof
   let X be non-empty 1-element FinSequence,
       S be SemialgebraFamily of X;
   set S1 = the set of all product <*s*> where s is Element of S.1;
   set X1 = the set of all <*x*> where x is Element of X.1;
A1:1 in Seg 1; then
   X.1 in S.1 by Th11; then
A3:product <*X.1*> in S1;
   S1 is cap-closed semiring_of_sets of X1 by SRINGS_4:34; then
A5:S1 is with_empty_element semi-diff-closed cap-closed Subset-Family of X1
     by SRINGS_3:10;
   1 in dom X by A1,FINSEQ_1:89; then
   product <*X.1*> = the set of all <*y*> where y is Element of X.1
     by SRINGS_4:24;
   hence thesis by A3,A5,SRINGS_3:def 6;
end;
