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) $f_{i}(a_0,\dots,a_{n_i-1}) \notin M$;
(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 $P$?