
theorem
  for X being non empty set,
      F being Point of C_Normed_Algebra_of_BoundedFunctions(X) holds
          Mult_(ComplexBoundedFunctions(X),CAlgebra(X)).(1r,F) = F
proof
  let X be non empty set,
      F be Point of C_Normed_Algebra_of_BoundedFunctions(X);
  set X1 = ComplexBoundedFunctions(X);
  reconsider f1 = F as Element of ComplexBoundedFunctions(X);
A1: [1r,f1] in [:COMPLEX,(ComplexBoundedFunctions X):];
  Mult_ (ComplexBoundedFunctions(X),CAlgebra(X)).(1r,F)
   =((the Mult of CAlgebra(X)) | [:COMPLEX,ComplexBoundedFunctions(X):]).
                                     (1r,f1) by Def3
  .= (the Mult of CAlgebra(X)).(1r,f1) by A1,FUNCT_1:49
  .= F by CFUNCDOM:12;
  hence thesis;
end;
