theorem Th24:
  R_Normed_Algebra_of_BoundedFunctions X is RealLinearSpace
proof
  (for v being VECTOR of R_Normed_Algebra_of_BoundedFunctions X holds 1 *
  v = v)& R_Normed_Algebra_of_BoundedFunctions X is Algebra by Th22,Th23;
  hence thesis by Lm2;
end;
