
theorem
  for X be non empty TopSpace
  for x be set st x in CC_0_Functions(X) holds
    x in CContinuousFunctions X
proof
  let X be non empty TopSpace;
  let x be set such that
A1: x in CC_0_Functions(X);
  consider f be Function of the carrier of X,COMPLEX such that
A2:        f=x & f is continuous
           & (ex Y be non empty Subset of X st Y is compact
           & (for A being Subset of X st A=support(f)
                          holds Cl(A) is Subset of Y)) by A1;
  thus thesis by A2;
end;
