reserve i,j for Nat;

theorem Th61:
  for V1 being Ring,P1,P2 be FinSequence of V1 st len P1 = len P2
  holds Sum(P1+P2) = (Sum P1) + (Sum P2)
proof
  let V1 be Ring, 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;
