
theorem
  for X being compact non empty TopSpace holds
  (R_Algebra_of_ContinuousFunctions(X) is
    Subalgebra of R_Algebra_of_BoundedFunctions the carrier of X)
proof
  let X be compact non empty TopSpace;
A1:the carrier of R_Algebra_of_ContinuousFunctions(X)
    c= the carrier of R_Algebra_of_BoundedFunctions the carrier of X
  by Lm1;
A2:R_Algebra_of_ContinuousFunctions(X) is
    Subalgebra of RAlgebra (the carrier of X) by C0SP1:6;
  R_Algebra_of_BoundedFunctions the carrier of X is
    Subalgebra of RAlgebra (the carrier of X) by C0SP1:10;
  hence thesis by A1,A2,Th8;
end;
