Proof
(related to Proposition: Uniqueness of Real Zero)
 We have to show that the real number zero \(0\) is unique, i.e. there can be only one such number, for which
\[x=x+0\quad\quad ( * )\]
for all $x\in\mathbb R$.
 Suppose, \(0^{\ast}\) is any (other) real number, for which
\[x=x+0^{\ast}\quad\quad ( * * )\]
for all $x\in\mathbb R$.
 Applying the commutativity law for adding real numbers, we get
\[\begin{array}{rcll}
0^{\ast}&=&0^{\ast}+0&\text{ by }( * )\\
&=&0+0^{\ast}&\text{ by commutativity of adding real numbers}\\
&=&0&\text{ by }( * * )
\end{array}
\]
 Thus, both zeros are in fact equal.
 Thus, the real zero $0$ is unique.
∎
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References
Bibliography
 Forster Otto: "Analysis 1, Differential und Integralrechnung einer VerĂ¤nderlichen", Vieweg Studium, 1983