
theorem
for X being non empty compact TopSpace holds
  C_Algebra_of_ContinuousFunctions X is
    ComplexSubAlgebra of C_Algebra_of_BoundedFunctions the carrier of X
proof
  let X be non empty compact TopSpace;
A1:the carrier of (C_Algebra_of_ContinuousFunctions X)
     c= the carrier of (C_Algebra_of_BoundedFunctions the carrier of X)
  by Lm1;
A2:C_Algebra_of_ContinuousFunctions X
    is ComplexSubAlgebra of CAlgebra the carrier of X by CC0SP1:2;
  C_Algebra_of_BoundedFunctions the carrier of X
    is ComplexSubAlgebra of CAlgebra the carrier of X by CC0SP1:4;
  hence C_Algebra_of_ContinuousFunctions X
    is ComplexSubAlgebra of C_Algebra_of_BoundedFunctions the carrier of X
                                   by A1,A2,Th15;
end;
