
theorem
  for L being left_zeroed add-associative non empty doubleLoopStr
  for B1,B2 being bag of the carrier of L
  for E,F being (the carrier of L)-valued FinSequence
  holds (B1+B2)(++)(E^F) = (B1(++)E)^(B1(++)F) + (B2(++)E)^(B2(++)F)
  proof
    let L be left_zeroed add-associative non empty doubleLoopStr;
    let B1,B2 be bag of the carrier of L;
    let E,F be (the carrier of L)-valued FinSequence;
    thus (B1+B2)(++)(E^F) = (B1(++)(E^F)) + (B2(++)(E^F)) by Th19
    .= (B1(++)E)^(B1(++)F) + (B2(++)(E^F)) by Th20
    .= (B1(++)E)^(B1(++)F) + (B2(++)E)^(B2(++)F) by Th20;
  end;
