What's the expression of "necessarily satisfied" in mathmatics?

The use of satisfied in this context is confusing.

The entire sequence being referenced is called a proof. The final statement in a proof is called a theorem:

[Merriam-Webster]
1 : a formula, proposition, or statement in mathematics or logic deduced or to be deduced from other formulas or propositions

From Wikipedia (additional emphasis mine):

In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the sequence by a rule of inference. It differs from a natural language argument in that it is rigorous, unambiguous and mechanically checkable …

The theorem is a syntactic consequence of all the well-formed formulas preceding it in the proof.

If anything is being satisfied in the proof, it's the set of axioms that precedes the theorem.

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However, you can say that the theorem is a necessary conclusion or a necessary consequence (of the set of axioms):

[Merriam-Webster]

conclusion
1 a : a reasoned judgment : INFERENCE
       // The obvious conclusion is that she was negligent.
1 b : the necessary consequence of two or more propositions taken as premises
especially : the inferred proposition of a syllogism

consequence
1 : a conclusion derived through logic : INFERENCE
    //… we can deduce … many consequences each of which can be tested by experiment.
— James Bryant Conant