We present a short proof of the following known result. Suppose $X, Y$ are finite connected CW-complexes with free involutions, $f colon X o Y$ is an equivariant map, and $l$ is a non-negative integer. If $f^* colon H^i (Y) o H^i (X)$ is an isomorphism for each $i>l$ and is onto for $i=l$, then $f^{sharp} colon pi^i_{eq}(Y)o pi^i_{eq}(X)$ is a $mbox{1-1}$ correspondence for $i>l$ and is onto for $i=l$.
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机译：我们提供以下已知结果的简短证明。假设$ X，Y $是具有自由对合的有限连通CW络合物，$ f 冒号X to Y $是等变映射，而$ l $是非负整数。如果$ f ^ * 冒号H ^ i（Y） to H ^ i（X）$是每个$ i> l $的同构并且在$ i = l $上，则$ f ^ { sharp} 冒号 pi ^ i_ {eq}（Y）到 pi ^ i_ {eq}（X）$是$ i> l $的$ mbox {1-1} $对应关系，而$ i = l $。
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