The statement $p:$ For any real numbers $x, y$ if $x=y$, then $2 x+a=2 y+a$ when $a \in \mathrm{Z}$.

We prove the statement $p$ is true by contrapositive method and by direct method. Direct method: For any real number $x$ and $y$, $\begin{array}{l} x=y \\ \Rightarrow 2 x=2 y \\ \Rightarrow 2 x+a=2 y+a \text { a for some } a \in Z \end{array}$ Contrapositive method: The contrapositive statement of $p$ is "For any real numbers $x, y$ if $2 \mathrm{x}+\mathrm{a} \neq 2 \mathrm{y}+\mathrm{a}$, where $\mathrm{a} \in \mathrm{Z}$, then $\mathrm{x} \neq \mathrm{y}$." Given, $2 x+a \neq 2 y+a$ $\begin{array}{l} \Rightarrow 2 x \neq 2 y \\ \Rightarrow x \neq y \end{array}$ Hence, the given statement is true.