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;

theorem Th15:
  0.R_Algebra_of_BoundedFunctions X = X -->0
proof
  R_Algebra_of_BoundedFunctions X is Subalgebra of RAlgebra X & 0.RAlgebra
  X = X -->0 by Th6;
  hence thesis by Th8;
end;
