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;
