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

theorem
  for S being ClassicalSemiringFamily of X holds
  MeasurableRectangle(S) is
  with_empty_element semi-diff-closed cap-closed Subset-Family of product X
  proof
    let S be ClassicalSemiringFamily of X;
    reconsider S1=S as cap-closed-yielding SemiringFamily of X by Thm38;
    SemiringProduct(S1) is cap-closed with_empty_element
    cap-finite-partition-closed diff-finite-partition-closed
    Subset-Family of product X by Thm32;
    hence thesis by SRINGS_3:10;
  end;
