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;
