Proof
(related to Proposition: Composition of Bijective Functions is Bijective)
 By hypothesis, Let $f:A\to B$ and $g:B\to C$ are bijective.
 Since $f,g$ are bijective, $f,g$ are in particular injective.
 By the corresponding proposition, the composition $(g\circ f)$ is also injective.
 Since $f,g$ are bijective, $f,g$ are in particular surjective.
 By the corresponding proposition, the composition $(g\circ f)$ is also surjective.
 Altogether, the composition $(g\circ f)$ is bijective.
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References
Bibliography
 Kane, Jonathan: "Writing Proofs in Analysis", Springer, 2016