
theorem Th3:
  for L being left_zeroed non empty addLoopStr, a being Element
  of L holds Sum <* a *> = a
proof
  let V be left_zeroed non empty addLoopStr, v be Element of V;
  reconsider a = v as Element of V;
  set S = <* v *>;
  consider f being sequence of  the carrier of V such that
A1: Sum(S) = f.(len S) and
A2: f.0 = 0.V & for j being Nat for v being Element of V st j
  < len S & v = S.(j + 1) holds f.(j + 1) = f.j + v by RLVECT_1:def 12;
A3: len <* a *> = 1 by FINSEQ_1:39;
  0 < 1;
  then consider j being Element of NAT such that
A4: j < len S;
A5: S.(j + 1) = S.(0 + 1) by A3,A4,NAT_1:14
    .= v;
  j = 0 by A3,A4,NAT_1:14;
  then f.1 = 0.V + v by A2,A5
    .= a by ALGSTR_1:def 2;
  hence thesis by A1,FINSEQ_1:39;
end;
