theorem Th12:
  F is with_the_same_dom & D c= dom(F.0) & x in D implies (
  Partial_Sums(F#x)).n = ((Partial_Sums F)#x).n
proof
  assume F is with_the_same_dom & D c= dom(F.0) & x in D;
  then (Partial_Sums((R_EAL F)#x)).n = ((Partial_Sums(R_EAL F))#x).n by Th9,
MESFUNC9:32;
  then (Partial_Sums(R_EAL(F#x))).n = ((Partial_Sums(R_EAL F))#x).n by
MESFUN7C:1;
  then (R_EAL(Partial_Sums(F#x))).n = ((Partial_Sums(R_EAL F))#x).n by Th5;
  then ((Partial_Sums(F#x)).n) = ((R_EAL(Partial_Sums F))#x).n by Th7;
  hence thesis by MESFUN7C:1;
end;
