A remarkable result of I. Shemer [4] states that the combinatorial
structure of a neighbourly $2m$-polytope determines the combinatorial
structure of each of its subpolytopes. From this, it follows that every
subpolytope of a cyclic $2m$-polytope is cyclic. In this note, we present
a direct proof of this consequence that also yields that certain
subpolytopes of a cyclic $(2m+1)$-polytope are cyclic.
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