A subalgebra of $P(M, \Delta)$ is a Peano Algebra

A subalgebra of $P(M, \Delta)$ is a Peano Algebra

Let $M$ be any set and let $\Delta=(n_i)_{i \in I}$ be an algebraic type.
Let $P=P(M, \Delta)$ be the algebra such that the generalized Peano Axioms
hold:
(P1) $f_{i}(a_0,\dots,a_{n-1}) \notin M$
(P2)$ f_{i}(a_0,\dots,a_{n_{i}-1}) = f_{j}(b_0,...,b_{n_{j}-1})$ only if
i=j and $a_{k}=b_{k}$ for all $k$.
(P3) the set $M$ generates $P$
The question is to prove that a subalgebra of $P(M, \Delta)$ is also a
Peano Algebra. Would the correct way to approach this problem is to just
check if all the axioms hold for any subalgebra of $P$?