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This key result justifies the establishment of sets’ magnitudes (cardinalities) by means of one-to-one correspondences, for it states that if, on that basis, the magnitudes of two sets are each no greater than the other, then the two magnitudes are equal. This theorem was conjectured by Georg Cantor (1845-1918). Proofs were published by Felix Bernstein and Ernst Schröder in the 1890s. Richard Dedekind had already proved this in 1887, but his proof was unpublished until the appearance of his collected works in 1932. The attached article (pdf) shows a proof I learned in graduate school.