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I'm new to everything involving languages and I'm having trouble proving properties with quantifier elimination. In particular what's below.

Suppose we know the following:

$Lemma$ (L1): Let $N \leq M$ be a substructure. Then for every quantifier-free $L$- formula ($L$ the language in this context) $\phi$ there is: $\phi$ holds in $N$ $\iff$ $\phi$ holds in $M$

Theorem (to prove): Let $L$ be a language, $T$ an $L$-theory, $M$ a model of $T$ and $A$ a substructure of $M$. Assume the theory admits quantifier elimination.

  1. Show that for every $L_A$-sentence $\phi$ there is an $L_A$-sentence $\psi$ such that: $\phi$ holds in $M$ $\iff$ $\psi$ holds in $A$.
  2. Suppose $M_1$ and $M_2$ are models of $T$, and $A$ is a substructure of both models. Show that $M_1$ and $M_2$ satisfy the same $L_A$-sentences.

I've tried proving 1. as follows:

Suppose $\phi$ holds in $M$, then there is a qf equivalent $\psi$ that holds in $M$. Becuase of (L1) $\psi$ holds in $M$ $\iff$ $\psi$ holds in $A$. $\blacksquare$

I'm not sure wether this is entirely correct, I feel like I'm missing something.

For 2. I've done (so far):

Take $\psi$ that holds in $A$, then:

there is $\phi$ that holds in $M_1$ because of (1); there is $\chi$ that holds in $M_2$ because of(1). and because of (1) if we take every $L_A$-sentence that holds in $A$ we can find every $L_A$-sentence that holds in $M_1$ and also every $L_A$-sentence that holds in $M_2$. So there are as many $L_A$-sentences in both models. I don't know how to prove they are the same though...

In general: I've never been the best at logics, so correct me if I do important things wrong and if I skip important steps because I 'assume' something is known.

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Your answer to $(1)$ is essentially right (I would only suggest to pay more attention to the fact that the formula $\phi$ is in $L_A$ and not in $L$).

Your answer to $(2)$ need some reworking, in my opinion. You can proceed as follows: let $\phi$ be an $L_A$-formula that holds in $M_1$, then by $(1)$ there is an equivalent $\psi$ without quantifiers. Then you can use $L1$ to say that $\psi$ holds true in $M_2$, and since $M_2\models T$ you can conclude that $M_2\models \phi$. This proves that $M_1\models \phi$ implies $M_2\models\phi$. Then, in an entirely similar way, you can go the other way around: take a formula $\phi$ true in $M_2$ and show that it is true in $M_1$.

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