For a short exact sequence of modules with projective resolutions of its outer terms, there is a projective resolution of the middle term forming a degreewise split short exact sequence of resolutions. Lift the degree-zero projective cover of the quotient into the middle module and add the projective cover of the submodule; this surjects onto the middle module. Its kernel fits into the corresponding short exact sequence of kernels. Repeating this construction proves the claim. Additive contravariant functors preserve the degreewise split exact sequences with their direction reversed.
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The Horseshoe Lemma is a result in topology and functional analysis, specifically in the context of the study of topological vector spaces. It is particularly important in the field of functional analysis, where it has applications in various areas including the theory of nonlinear operators and differential equations. The lemma generally states that under certain conditions, a continuous linear operator defined on a Banach space can be approximated by finite-dimensional spaces in a specific way.