The two-column format is the method by which many students are introduced to formal proof-writing in mathematics. The student divides the page into two columns. In the left column goes a list of statements, each one a consequence of the one above it in the list. Adjacent to each such statement (in the right column) is the reason why this statement does indeed follow from the previous step. Eventually, some non-trivial result is obtained.
Proofs in higher levels of mathematics drop the explicit two-column format, but may still employ the method implicitly. For example, if a derivation requires a step which is not obvious, the proof writer may quote the source of the result, and this will appear immediately to the right of the statement in question. This may involve putting a reference to the original paper in which the result appeared, or else the name of a theorem or lemma which is widely known in the literature. (One might say such-and-such-a-step follows "by Caratheodory's theorem", for instance.)
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