Poizat's definition of $p$-equivalent $k$-tuples Maybe this is too trivial a question to be posted anywhere, but anyway. I am reading Poizat's "A Course in Model Theory". In  page 4  he defines the notion of two $k$-tuples, each in the universe of some relation, being $p$-equivalent.
He gives two conditions:


*

*$a_i = a_j \leftrightarrow b_i = b_j$.

*The function $s$ defined by $sa_1=b_1,...,sa_k=b_k$ is a $p$-isomorphism from $R$ to $R'$.


I guess the first of them is just a remark, because it seems redundant given the second. If the function $s$ is a $p$-isomorphism then it is, in particular, a bijection between the finite subsets $\{a_1,...,a_k\}$ and $\{b_1,...,b_k\}$, and hence it is impossible to falsify condition 1. Am I missing anything?

Yes, I was missing something pretty obvious. Condition 1 is what enables him to talk about the function $s$ in condition 2. Otherwise he should have talked about the relation $s$.
