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 Th36:
  R_Normed_Algebra_of_BoundedFunctions X is RealBanachSpace
proof
  for seq be sequence of R_Normed_Algebra_of_BoundedFunctions X st seq is
  Cauchy_sequence_by_Norm holds seq is convergent by Th35;
  hence thesis by LOPBAN_1:def 15;
end;
