 reserve
  S for non empty TopSpace,
  T for LinearTopSpace,
  X for non empty Subset of the carrier of S;
 reserve
    S,T for RealNormSpace,
    X for non empty Subset of the carrier of S;

theorem
  for X be non empty TopSpace, T be NormedLinearTopSpace
  for x being set st x in C_0_Functions(X,T) holds
  x in ContinuousFunctions(X,T)
  proof
    let X be non empty TopSpace,T be NormedLinearTopSpace;
    let x be set;
    assume x in C_0_Functions(X,T); then
    consider f be Function of the carrier of X, the carrier of T such that
A2: f=x & f is continuous
    & ex Y be non empty Subset of X st Y is compact
    & Cl(support(f)) c= Y;
    thus thesis by A2;
  end;
