
theorem
  for X being non empty TopSpace
  for W be non empty Subset of CAlgebra the carrier of X
                                                 st W= CC_0_Functions X
         holds W is Cadditively-linearly-closed
proof
  let X being non empty TopSpace;
  let W be non empty Subset of CAlgebra the carrier of X;
  assume
A1: W= CC_0_Functions X;
  set V = CAlgebra the carrier of X;
  for v,u be Element of V st v in W & u in W holds v + u in W by A1,Lm10;
  then
A2: W is add-closed by IDEAL_1:def 1;
  for v be Element of V st v in W holds -v in W by A1,Lm12;
  then
A3: W is having-inverse by C0SP1:def 1;
  for a be Complex, u be Element of V st u in W holds a * u in W
                                                            by A1,Lm11;
  hence W is Cadditively-linearly-closed  by A2,A3;
end;
