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 Th17:
  for X be non empty set, F be Functional_Sequence of X,REAL st F
  is with_the_same_dom holds Partial_Sums F is with_the_same_dom
proof
  let X be non empty set, F be Functional_Sequence of X,REAL;
  assume
A1: F is with_the_same_dom;
  let n,m be Nat;
  dom((Partial_Sums F).n) = dom(F.0) by A1,Th11;
  hence dom((Partial_Sums F).n) = dom((Partial_Sums F).m) by A1,Th11;
end;
