(related to Proposition: Inequality of Natural Numbers and Their Successors)
According to the definition of natural numbers, they fulfill the Peano axioms. By virtue of the axiom P4 we have
By virtue of the proving principle by contraposition it follows
\[x\neq y\Longrightarrow x^+\neq y^+,\]
i.e. if any two natural numbers are unequal, then their successors are also unequal.
- Landau, Edmund: "Grundlagen der Analysis", Heldermann Verlag, 2008