Method of the Auxiliary Hypothesis

Definition
The method rests on the Criterion of Deduction:

Let $$A$$ be a relation in $$I$$, and $$I*$$ be the the theory obtained by adjoining $$A$$ to the axioms of $$I$$. If $$B$$ is a theorem in $$I*$$, then $$A \Rightarrow B$$ is a theorem in $$I$$.

The proof is on the page 30 in the book Theory of Sets by Bourbaki.

Example
Suppose a field, where the inverse of the zero-element and its existence are not defined in axioms.

'''The auxiliary hypothesis: ''' adjoin the inconsistent assumption (not yet proved) to the axioms that the zero-element has an inverse:

$${0}^{-1}*0 = 1$$

According to the axioms of the field:

$$0^{-1}*0 = 0*0^{-1}= 0$$

Contradiction, since an axiom of the field states the uniqueness of the product. So the zero-element has no inverse by the Method of the auxiliary hypothesis. Otherwise, the set of axioms would be inconsistent.

A common mistake is to assume the non-existence of the inverse for the zero-element from the axioms of the field. It is a consequence, not a definition per se.

QED

Similarity to a proof method from Propositional Logic
If $$a \and \neg b$$ leads to contradiction, then the proposition $$(a\rightarrow b)$$ is right.

$$(a\rightarrow b) \Leftrightarrow (\neg a \or b) \Leftrightarrow \neg (a \and \neg b)$$