theorem Th30:
  for P1,P2 being FinSequence of V1 st len P1 = len P2 holds
  Sum(P1 + P2) = (Sum P1) + (Sum P2)
  proof
    let P1, P2 be FinSequence of V1;
    assume len P1 = len P2;
    then reconsider R1 = P1, R2 = P2
    as Element of (len P1) -tuples_on the carrier of V1 by FINSEQ_2:92;
    thus Sum (P1 + P2) = Sum (R1 + R2)
    .= (Sum P1) + (Sum P2) by FVSUM_1:76;
  end;
