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;
reserve F for Functional_Sequence of X,COMPLEX,
  f for PartFunc of X,COMPLEX,
  A for set;

theorem Th34:
  F is with_the_same_dom implies Partial_Sums F is with_the_same_dom
proof
  assume F is with_the_same_dom;
  then Re F is with_the_same_dom;
  then Partial_Sums Re F is with_the_same_dom by Th17;
  then Re(Partial_Sums F) is with_the_same_dom by Th29;
  hence Partial_Sums F is with_the_same_dom by Th24;
end;
