theorem
  z in dom((Partial_Sums F).n) & m <= n implies z in dom((Partial_Sums F
  ).m) & z in dom(F.m)
proof
  assume
A1: z in dom((Partial_Sums F).n) & m <= n;
A2: dom((Partial_Sums F).n) = dom((Re(Partial_Sums F)).n) by MESFUN7C:def 11
    .= dom((Partial_Sums Re F).n) by Th29;
  dom((Partial_Sums Re F).m) = dom((Re(Partial_Sums F)).m) by Th29
    .= dom((Partial_Sums F).m) by MESFUN7C:def 11;
  hence z in dom((Partial_Sums F).m) by A1,A2,Th8;
  z in dom((Re F).m) by A1,A2,Th8;
  hence z in dom(F.m) by MESFUN7C:def 11;
end;
