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Zorn's lemma states that a nonempty partially ordered set in which every chain has an upper bound has a maximal element. Order the partial group operations by inclusion of their graphs. The collection is a set, since its members are subsets of $X\times X\times X$. It is nonempty: choose $x_0\in X$ and put the one-element group operation on $\{x_0\}$.

To construct a chain union of partial group operations, for the given nonempty chain, let $m={\bigcup }_{n\in \mathcal{C}}n$ and let $A$ be the union of the corresponding group carriers. Comparability guarantees agreement wherever two operations are both defined, so $m$ is a function. Its domain is $A\times A$: any two elements of $A$ lie together in one carrier, by taking the larger of the two chain members containing them. Their product then lies in that carrier and hence in $A$.

Any three elements similarly lie together in one chain member, where associativity holds; therefore associativity holds for $m$. The identities of nested nonempty groups agree: the smaller identity remains an idempotent under the larger group operation, whose only idempotent is its identity. Hence the whole chain has a common identity. Each element's inverse in a carrier remains its inverse in every larger carrier, so it is an inverse in the union as well. Thus $(A,m)$ satisfies all group axioms andIt is an upper bound for the chain. The empty chain also has an upper bound because $\mathcal{G}$ is nonempty, completing the hypothesis needed for Zorn's lemma.