Extension of Contraction of Extension of Ideal is Extension

Theorem
Let $A$ and $B$ be commutative rings with unity.

Let $f : A \to B$ be a ring homomorphism.

Let $\mathfrak a$ be an ideal of $A$.

Let ${\mathfrak a}^e$ be the extension of $\mathfrak a$ by $f$.

Let ${\mathfrak a}^{ec}$ be the contraction of ${\mathfrak a}^e$ by $f$.

Let ${\mathfrak a}^{ece}$ be the extension of ${\mathfrak a}^{ec}$ by $f$.

Then:
 * ${\mathfrak a}^e = {\mathfrak a}^{ece}$

Proof
Since $\mathfrak a \subseteq {\mathfrak a}^{ec}$ by Ideal is Contained in Contraction of Extension: