
theorem Th1:
  for X be add-associative right_zeroed right_complementable non
empty CNORMSTR for seq be sequence of X st (for n be Nat holds seq.
  n = 0.X) holds for m be Nat holds (Partial_Sums seq).m = 0.X
proof
  let X be add-associative right_zeroed right_complementable non empty
  CNORMSTR;
  let seq be sequence of X such that
A1: for n be Nat holds seq.n = 0.X;
  let m be Nat;
  defpred P[Nat] means seq.$1 = (Partial_Sums seq).$1;
A2: for k be Nat st P[k] holds P[k+1]
  proof
    let k be Nat such that
A3: P[k];
    thus seq.(k+1) = 0.X + seq.(k+1) by RLVECT_1:4
      .= (Partial_Sums seq).k + seq.(k+1) by A1,A3
      .= (Partial_Sums seq).(k+1) by BHSP_4:def 1;
  end;
A4: P[0] by BHSP_4:def 1;
  for n be Nat holds P[n] from NAT_1:sch 2(A4,A2);
  then for n be Element of NAT holds P[n];
  then seq = Partial_Sums seq by FUNCT_2:63;
  hence thesis by A1;
end;
