
theorem Th17:
  for X being non empty set holds
    C_Normed_Algebra_of_BoundedFunctions(X) is ComplexLinearSpace
proof
  let X be non empty set;
  for v being Point of C_Normed_Algebra_of_BoundedFunctions(X)
                                        holds 1r*v = v
  proof
    let v be Point of C_Normed_Algebra_of_BoundedFunctions(X);
    reconsider v1 = v as Element of ComplexBoundedFunctions(X);
A1: 1r*v = ((the Mult of CAlgebra(X))|
             [:COMPLEX,ComplexBoundedFunctions(X):]).([1r,v1]) by Def3
        .= (the Mult of CAlgebra(X)).(1r,v1) by FUNCT_1:49
        .= v1  by CFUNCDOM:12;
    thus thesis by A1;
  end;
  hence thesis by Th15,CLVECT_1:def 5;
end;
