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

theorem Thm38:
  for S being ClassicalSemiringFamily of X holds
  S is cap-closed-yielding SemiringFamily of X
  proof
    let S be ClassicalSemiringFamily of X;
    now
      let i be Nat;
      assume
A1:   i in Seg n;
      S.i is with_empty_element semi-diff-closed cap-closed
      Subset-Family of X.i by A1,Def5;
      hence S.i is semiring_of_sets of X.i by SRINGS_3:9;
    end;
    then reconsider SC=S as SemiringFamily of X by Def2;
    for i be Nat st i in Seg n holds SC.i is cap-closed by Def5;
    hence thesis by Def4;
  end;
