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 Th32:
  F is with_the_same_dom implies dom((Partial_Sums F).n) = dom(F.0 )
proof
  assume F is with_the_same_dom;
  then Re F is with_the_same_dom;
  then dom((Partial_Sums Re F).n) = dom((Re F).0) by Th11;
  then dom((Partial_Sums Re F).n) = dom(F.0) by MESFUN7C:def 11;
  then dom((Re Partial_Sums F).n) = dom(F.0) by Th29;
  hence dom((Partial_Sums F).n) = dom(F.0) by MESFUN7C:def 11;
end;
