It's just a way of breaking down the full proof into pieces.

Lemma 2.1 says 'if this assignment exists then X'

Then later in the proof you say 'here is such an assignment, so, applying lemma 2.1, therefore X'

You don't need to assume the existence of the assignment, you prove that if the assignment exists then something else follows, and then later if you can find that assignment then you get the result of lemma 2.1.