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;
