I have some problems with the following theorem: Fix an signature $\sigma$ and a set of variables $\mathbb{V}$. We call $t$ and $t_1$ "equivalent", if for every $\sigma$-structure $S$ and every term function $\beta: T \rightarrow \underline{S}$, where $T$ is the set of all $\sigma$-terms and $\underline{S}$ the underlying set of the structure $S$, one has $\beta(t)=\beta(t_1)$. One has then to show, that $t$ is equivalent to $t_1$ iff $t=t_1$.
EDIT: The "counterexample" I previously provided was incorrect, since it wasn't a proper counterexample. (So in terms of what I previously wrote here, the theorem seems to be correct). I still would like to have a full proof for it. The idea for the nontrivial direction ("$\Rightarrow$") is to use the term algebra. My idea was that roughly that I since I know that in a term algebra $\beta(t)=\beta(t_1) \Leftrightarrow t=t_1$ trivially holds, since it is an obvious tautology, I think I should somehow show that every other structure can somehow transformed into a term algebra. Has someone some ideas how to do this ?