reserve X for non empty set;
reserve x for Element of X;
reserve d1,d2 for Element of X;
reserve A for BinOp of X;
reserve M for Function of [:X,X:],X;
reserve V for Ring;
reserve V1 for Subset of V;
reserve V for Algebra;
reserve V1 for Subset of V;
reserve MR for Function of [:REAL,X:],X;
reserve a for Real;
reserve F,G,H for VECTOR of R_Algebra_of_BoundedFunctions X;
reserve f,g,h for Function of X,REAL;
reserve F,G,H for Point of R_Normed_Algebra_of_BoundedFunctions X;

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;
