
theorem Th1:
  for V being add-associative left_zeroed right_zeroed non empty
  addLoopStr, v,u being Element of V holds Sum <* v,u *> = v + u
proof
  let V be add-associative left_zeroed right_zeroed non empty addLoopStr, v,
  u be Element of V;
  <* v,u *> = <* v *> ^ <* u *> by FINSEQ_1:def 9;
  then Sum<* v,u *> = Sum<* v *> + Sum<* u *> by RLVECT_1:41
    .= v + Sum <* u *> by BINOM:3
    .= v + u by BINOM:3;
  hence thesis;
end;
