
theorem
  for X be non empty TopSpace
  for W be non empty Subset of RAlgebra the carrier of X
      st W = C_0_Functions(X) holds W is additively-linearly-closed
proof
  let X be non empty TopSpace;
  let W be non empty Subset of RAlgebra the carrier of X;
  assume
A1: W = C_0_Functions(X);
  set V = RAlgebra 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;
  for a be Real, u be Element of V st u in W holds a * u in W by A1,Lm11;
  hence W is additively-linearly-closed by A2,A3;
end;
