If we forget about for a moment, what's the use to selectsomeand such that implies ? This implication holds foralland . Also, how does it help to prove ? You need toassumethis for some and and show that .

Each structure provides an interpretation for some language. If a constant is not in the language of structure and , then by I'll denote a structure for the same language with added which maps into . Use the following facts where does not occur in , , or the language of .

Lemma 1. iff .

Lemma 2. iff .