reserve X for set,
  F for Field_Subset of X,
  M for Measure of F,
  A,B for Subset of X,
  Sets for SetSequence of X,
  seq,seq1,seq2 for ExtREAL_sequence,
  n,k for Nat;

theorem Th1:
  Ser seq = Partial_Sums seq
proof
  for x be object st x in NAT holds Ser(seq).x = (Partial_Sums seq).x
  proof
    defpred P[Nat] means Ser(seq).$1 = (Partial_Sums seq).$1;
    let x be object;
A1: for k be Nat st P[k] holds P[k+1]
    proof
      let k be Nat;
      assume P[k];
      then (Ser seq).(k+1) = (Partial_Sums seq).k + seq.(k+1) by SUPINF_2:
        def 11;
      hence P[k+1] by MESFUNC9:def 1;
    end;
    assume x in NAT;
    then reconsider n = x as Element of NAT;
    (Ser seq).0 = seq.0 by SUPINF_2:def 11
      .= (Partial_Sums seq).0 by MESFUNC9:def 1;
    then
A2: P[0];
    for k be Nat holds P[k] from NAT_1:sch 2(A2,A1);
    then (Ser seq).x = (Partial_Sums seq).n;
    hence (Ser seq).x = (Partial_Sums seq).x;
  end;
  hence Ser seq = Partial_Sums seq by FUNCT_2:12;
end;
