reserve X for RealNormSpace;

theorem Th1:
  for X be RealNormSpace,
      R1, R2, R3 be FinSequence of X
        st len R1 = len R2 & R3 = R1 + R2 holds Sum(R3) = Sum(R1) + Sum(R2)
proof
  let X be RealNormSpace,
      R1, R2, R3 be FinSequence of X;
  assume A1: len R1 = len R2 & R3 = R1 + R2; then
  dom R1 = dom R2 by FINSEQ_3:29;
  hence Sum(R3) = Sum(R1) + Sum(R2) by A1,BINOM:7;
end;
