Upwards Absolute

Let and be Structures for a Language such that
Let be a formula in Free Variables.
We say that is upwards absolute between and if

where we use to mean is Satisfied in

Lemma

Suppose formula is upwards absolute between and ,
and let be among the Free Variables of .
Then is upwards absolute.

Proof

The formula has Free Variables.
Suppose that for some :

i.e. there are some such that

By assumption, is upwards absolute so:

so indeed