theorem Th33:
  R_Normed_Algebra_of_BoundedFunctions X is
   reflexive discerning RealNormSpace-like
proof
 thus ||.0.R_Normed_Algebra_of_BoundedFunctions X.|| = 0 by Th32;
  for x, y being Point of R_Normed_Algebra_of_BoundedFunctions X for a be
Real
   holds ( ||.x.|| = 0 iff x = 0.R_Normed_Algebra_of_BoundedFunctions(X) ) &
  ||.a*x.|| = |.a.| * ||.x.|| & ||.x+y.|| <= ||.x.|| + ||.y.|| by Th32;
  hence thesis by NORMSP_1:def 1;
end;
