Proof. The map φ preserves the operations of addition and multiplication due to
and
Furthermore the additive and multiplicative identities of are mapped to the additive and multiplicative
identities of ℝ:
and
__
With this definition of φ, we have that the kernel is exactly I = {f ∈ | f(c) = 0}. Hence with the above lemma, I
is an ideal of
. Futhermore, due to the above lemma and because φ is surjective (for a ∈ [0,1], the map x
a is
mapped to a by φ), the Fundamental Theorem of Ring Homomorphisms implies that
∕I is isomorphic to ℝ since ℝ
is a field, this implies that the ideal I is a maximal ideal.