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

theorem Thm33:
  for X being non-empty 1-element FinSequence,
  S being cap-closed-yielding SemiringFamily of X holds
  the set of all product <*s*> where s is Element of S.1 is
  cap-closed semiring_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 cap-closed-yielding SemiringFamily of X;
    S is cap-closed-yielding SemiringFamily of X & 1 in Seg 1 by FINSEQ_1:3;
    then
A1: S.1 is semiring_of_sets of X.1 & S.1 is cap-closed by Def2,Def4;
    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;
    now
      let s1,s2 be set;
      assume that
A2:   s1 in S1 and
A3:   s2 in S1;
      consider t1 be Element of S.1 such that
A4:   s1 = product <*t1*> by A2;
      consider t2 be Element of S.1 such that
A5:   s2 = product <*t2*> by A3;
A6:   s1/\s2 = product <*t1/\t2*> by A4,A5,Thm25;
      t1/\t2 is Element of S.1 by A1;
      hence s1/\s2 in S1 by A6;
    end;
    then S1 is cap-closed;
    hence thesis by Thm27;
  end;
