Hey could somebody have a look at the following proof atempt and tell me if im on the right way ? Thanks in advance!
Claim: Let $M$ and $N$ be two structures and $a \in M$ and $b \in N$. TFAE.
1.) (a,b) is a partial isomorphism between $M$ and $N$.
2.) $tp(a) \cap QF = tp(b) \cap QF$
where QF is the set of quantifier free formulas.
$(1 \Rightarrow 2)$
Since we have a local ismorphism between $M$ and $N$ we know that $M$ and $N$ satisfy the same formulas, so by definition $ tp(a)=tp(q) $. Now since both models satisfy the same set of quantifier free formulas, we get immediately the desiderata.