reserve X for non empty set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  E for Element of S,
  F,G for Functional_Sequence of X,ExtREAL,
  I for ExtREAL_sequence,
  f,g for PartFunc of X,ExtREAL,
  seq, seq1, seq2 for ExtREAL_sequence,
  p for ExtReal,
  n,m for Nat,
  x for Element of X,
  z,D for set;

theorem Th32:
  F is additive & F is with_the_same_dom & D c= dom(F.0) & x in D
  implies (Partial_Sums(F#x)).n = ((Partial_Sums F)#x).n
proof
  set PF = Partial_Sums F;
  set PFx = Partial_Sums(F#x);
  assume that
A1: F is additive and
A2: F is with_the_same_dom and
A3: D c= dom(F.0) and
A4: x in D;
  defpred P[Nat] means PFx.$1 = (PF#x).$1;
A5: for k be Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    assume
A6: P[k];
    PFx.(k+1) = PFx.k + (F#x).(k+1) by Def1;
    then PFx.(k+1) = (PF#x).k + (F.(k+1)).x by A6,MESFUNC5:def 13;
    then
A7: PFx.(k+1) = (PF.k).x + (F.(k+1)).x by MESFUNC5:def 13;
A8: PF.(k+1) = PF.k + F.(k+1) by Def4;
    D c= dom(PF.(k+1)) by A1,A2,A3,Th29;
    then PFx.(k+1) = (PF.(k+1)).x by A4,A8,A7,MESFUNC1:def 3;
    hence thesis by MESFUNC5:def 13;
  end;
  PFx.0 = (F#x).0 by Def1;
  then PFx.0 = (F.0).x by MESFUNC5:def 13;
  then PFx.0 = (PF.0).x by Def4;
  then
A9: P[ 0 ] by MESFUNC5:def 13;
  for k being Nat holds P[k] from NAT_1:sch 2(A9,A5);
  hence thesis;
end;
