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 Th8:
  z in dom((Partial_Sums F).n) & m <= n implies z in dom((
  Partial_Sums F).m) & z in dom(F.m)
proof
  (Partial_Sums F).n = (R_EAL(Partial_Sums F)).n;
  then
A1: (Partial_Sums F).n = (Partial_Sums R_EAL F).n by Th7;
  (Partial_Sums R_EAL F).m = (R_EAL(Partial_Sums F)).m by Th7;
  hence thesis by A1,MESFUNC9:22;
end;
