 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 Th34:
  for S be non empty compact TopSpace,T be NormedLinearTopSpace holds
  for x being set st x in ContinuousFunctions(S,T) holds
  x in BoundedFunctions (the carrier of S,T)
proof
  let S be non empty compact TopSpace,T be NormedLinearTopSpace;
  let x be set;
  assume x in ContinuousFunctions(S,T); then
  consider f be Function of the carrier of S,the carrier of T such that
A1:x=f & f is continuous;
A2:dom f = the carrier of S by FUNCT_2:def 1;
  consider K be Real such that
A3: 0 <= K &
  for x being Element of S st x in [#]S
    holds ||.f. x.|| <= K by A1,A2,Lm2,COMPTS_1:1;
  for x being Element of S holds ||.f. x.|| <= K by A3; then
  f is bounded by RSSPACE4:def 4,A3;
  hence x in BoundedFunctions (the carrier of S,T) by RSSPACE4:def 5,A1;
end;
