We give a short proof of the fundamental theorem of central element theory (see: Sanchez Terraf and Vaggione, Varieties with definable factor congruences, T.A.M.S. 361). The original proof is constructive and very involved and relies strongly on the fact that the class be a variety. Here we give a more direct nonconstructive proof which applies for the more general case of a first-order class which is both closed under the formation of direct products and direct factors.

中文翻译：

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中心元论的基本定理

我们给出了中心元理论基本定理的简短证明（参见：Sanchez Terraf 和 Vaggione，具有可定义因子同余的品种，TAMS 361）。原始证明是建设性的并且非常复杂，并且强烈依赖于类是多样化的这一事实。在这里，我们给出了一个更直接的非建设性证明，它适用于更一般的一阶类情况，该类在直接产品和直接因素的形成下都是封闭的。