In statistics, there is the Fundamental Lemma of Neyman-Pearson (see, e.g. Erich Lehmann. Fisher, Neyman, and the creation of classical statistics; E. Lehmann & J. Romano. Testing statistical hypotheses, 3.2. The Neyman-Pearson Fundamental Lemma) though I fail to see how is it "fundamental". Moreover, in different books this lemma is formulated in different ways.