theorem Th43:
  for X be non empty set, F be Functional_Sequence of X,ExtREAL st
  F is additive & 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,ExtREAL;
  assume that
A1: F is additive and
A2: F is with_the_same_dom;
  let n,m be Nat;
  dom((Partial_Sums F).n) = dom(F.0) by A1,A2,Th29;
  hence thesis by A1,A2,Th29;
end;
