reserve X for non empty set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  E for Element of S,
  F for Functional_Sequence of X,REAL,

  f for PartFunc of X,REAL,
  seq for Real_Sequence,
  n,m for Nat,
  x for Element of X,
  z,D for set;
reserve i for Element of NAT;

theorem Th5:
  Partial_Sums R_EAL seq = R_EAL(Partial_Sums seq)
proof
  defpred P[Nat] means (Partial_Sums(R_EAL seq)).$1 = (R_EAL
  Partial_Sums(seq)).$1;
A1: for k be Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    assume P[k];
    then
    (Partial_Sums(R_EAL seq)).(k+1) = Partial_Sums(seq).k + seq
    .(k+1) by MESFUNC9:def 1
      .=Partial_Sums(seq).k + seq.(k+1);
    hence thesis by SERIES_1:def 1;
  end;
  (Partial_Sums(R_EAL seq)).0 = seq.0 by MESFUNC9:def 1;
  then
A2: P[ 0 ] by SERIES_1:def 1;
  for i being Nat holds P[i] from NAT_1:sch 2(A2,A1);
  hence thesis;
end;
