 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 Th60:
  for X be non empty TopSpace,T be NormedLinearTopSpace
  for x being set st x in C_0_Functions(X,T) holds
  x in BoundedFunctions(the carrier of 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;
  consider Y be non empty Subset of X such that
A3: Y is compact & Cl(support(f)) c= Y by A2;
A4: dom f = the carrier of X by FUNCT_2:def 1; then
  consider K be Real such that
A5: 0 <= K &
  for x be Point of X st x in Y holds ||.f.x.|| <= K by A3,A2,Lm2;
  for x being Element of X holds ||.f. x.|| <= K
  proof
    let x be Element of X;
A6: support(f) c= Cl support(f) by PRE_TOPC:18;
    per cases;
    suppose not x in Y; then
      not x in support(f) by A3,A6; then
      not x in dom f or f/.x = 0.T by Def10;
      hence ||.f. x.|| <= K by A5,A4;
    end;
    suppose x in Y;
      hence ||.f. x.|| <= K by A5;
    end;
  end; then
  f is bounded by RSSPACE4:def 4,A5;
  hence x in BoundedFunctions(the carrier of X,T) by RSSPACE4:def 5,A2;
end;
