(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 \[x^+=y^+\Longrightarrow x=y.\]
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.