Show an AEC with intersections is appropriately named

I've just started learning the basics about AECs from this set of lecture notes. I am asked to show that if $M$ has intersections then the arbitrary intersection $S = \cap{S_i}$ is a strong substructure of $M$ if each $S_i$ is. (Here $A$ is said to be a strong substructure of $M$ if $A \leq_K M$.)

I've tried looking at $Cl(\cap{S_i})$ and the only thing I can think of is to use coherence, but this doesn't quite get me there.

Solution 1:

Suppose $S = \bigcap_{i\in I} S_i$, where $S_i\leq_K N$ for all $i\in I$. Then $$S\subseteq \text{cl}^N(S) = \bigcap \{M\leq_K N\mid S\subseteq M\} \subseteq \bigcap_{i\in I} S_i = S.$$ Since $K$ has intersections, we have $S = \text{cl}^N(S) \leq_K N$.