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;
