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 Th22:
  z in dom((Partial_Sums F).n) & m <= n implies z in dom((
  Partial_Sums F).m) & z in dom(F.m)
proof
  set PF = Partial_Sums F;
  assume that
A1: z in dom(PF.n) and
A2: m <= n;
  thus
A3: z in dom(PF.m) by A1,A2,Lm1;
  per cases;
  suppose
    m = 0;
    then PF.m = F.m by Def4;
    hence thesis by A1,A2,Lm1;
  end;
  suppose
    m <> 0;
    then consider k be Nat such that
A4: m = k + 1 by NAT_1:6;
    PF.m = PF.k + F.m by A4,Def4;
    then dom(PF.m) = (dom(PF.k) /\ dom(F.m)) \(((PF.k)"{-infty} /\ (F.m)"{
    +infty}) \/ ((PF.k)"{+infty} /\ (F.m)"{-infty})) by MESFUNC1:def 3;
    then z in dom(PF.k) /\ dom(F.m) by A3,XBOOLE_0:def 5;
    hence thesis by XBOOLE_0:def 4;
  end;
end;
