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