theorem Th8:
  z in dom((Partial_Sums F).n) & m <= n implies z in dom((
  Partial_Sums F).m) & z in dom(F.m)
proof
  (Partial_Sums F).n = (R_EAL(Partial_Sums F)).n;
  then
A1: (Partial_Sums F).n = (Partial_Sums R_EAL F).n by Th7;
  (Partial_Sums R_EAL F).m = (R_EAL(Partial_Sums F)).m by Th7;
  hence thesis by A1,MESFUNC9:22;
end;
