theorem
  R_Algebra_of_BoundedFunctions X is RealLinearSpace
proof
  now
    let v being VECTOR of R_Algebra_of_BoundedFunctions X;
    reconsider v1=v as VECTOR of RAlgebra X by TARSKI:def 3;
    R_Algebra_of_BoundedFunctions X is Subalgebra of RAlgebra X by Th6;
    then jj * v = jj*v1 by Th8;
    hence 1 * v =v by FUNCSDOM:12;
  end;
  hence thesis by Lm2;
end;
