Proof

(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.


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References

Bibliography

  1. Landau, Edmund: "Grundlagen der Analysis", Heldermann Verlag, 2008