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 Th11:
  F is with_the_same_dom implies dom((Partial_Sums F).n) = dom(F.0 )
proof
  assume F is with_the_same_dom;
  then dom((Partial_Sums R_EAL F).n) = dom((R_EAL F).0) by Th9,MESFUNC9:29;
  then dom((R_EAL(Partial_Sums F)).n) = dom((R_EAL F).0) by Th7;
  hence thesis;
end;
