 Using this definition, Frege 1879 proves some non-trivial mathematical truths, such as that the ancestral \(\Rarel\) is transitive and that, for any functional relation \(R\), the \(R\)-ancestors of any object are \(\Rarel\)-comparable (that is, he proved: Functional \((R) \amp x\Rarel y \amp x\Rarel z \rightarrow y\Rarel z \lor z\Rarel y\)).