let X be non empty set ; :: thesis: R_Normed_Algebra_of_BoundedFunctions X is RealBanachSpace
for seq being sequence of (R_Normed_Algebra_of_BoundedFunctions X) st seq is Cauchy_sequence_by_Norm holds
seq is convergent by Th35;
hence R_Normed_Algebra_of_BoundedFunctions X is RealBanachSpace by LOPBAN_1:def 15; :: thesis: verum