The following is a quotation from the proof of Proposition 11.10 in "Introduction to Commutative Algebra" by Atiyah and MacDonald.
Also if ${\mathfrak m}'$ is the maximal ideal of $A'$,
$A'/{\mathfrak m}'^n$ is a homomorphic image of $A/{\mathfrak m}^n$, hence $l(A/{\mathfrak m}^n) \geq l(A'/{\mathfrak m}'^n)$.
In the above, $A$ is a Noetherian local ring, ${\mathfrak m}$ is its maximal ideal, and $A'=A/{\mathfrak p}_0$ where ${\mathfrak p}_0$ is a prime ideal in $A$. Also, $l(M)$ is the length of $M$.
It seems to me that
$A'/{\mathfrak m}'^n$ is an isomorphic image of $A/{\mathfrak m}^n$, hence $l(A/{\mathfrak m}^n) = l(A'/{\mathfrak m}'^n)$.
Am I wrong ?