
theorem Lm2:
  for A be non empty closed_interval Subset of REAL,
      v be Point of
        R_Normed_Algebra_of_ContinuousFunctions ClstoCmp(A)
  holds
    v in BoundedFunctions the carrier of ClstoCmp(A)
proof
  let A be non empty closed_interval Subset of REAL,
      v be Point of
        R_Normed_Algebra_of_ContinuousFunctions(ClstoCmp(A));
  set AV = the carrier of ClstoCmp(A);
  R_Algebra_of_ContinuousFunctions ClstoCmp(A) is
    Subalgebra of R_Algebra_of_BoundedFunctions(AV) by C0SP2:9; then
A1: the carrier of
      R_Algebra_of_ContinuousFunctions ClstoCmp(A) c=
      the carrier of
        R_Algebra_of_BoundedFunctions(AV) by C0SP1:def 9;
  v in the carrier of
    R_Algebra_of_ContinuousFunctions ClstoCmp(A);
  hence v in BoundedFunctions(AV) by A1;
end;
