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\begin{alignat}{2}
\exp\left(-\frac{1}{2}\|\mathbf{x} - \mathbf{x'}\|^2\right)
&= \exp(\frac{2}{2}\mathbf{x}^\top \mathbf{x'} - \frac{1}{2}\|\mathbf{x}\|^2 - \frac{1}{2}\|\mathbf{x'}\|^2)\\[5pt]
&= \exp(\mathbf{x}^\top \mathbf{x'}) \exp( - \frac{1}{2}\|\mathbf{x}\|^2) \exp( - \frac{1}{2}\|\mathbf{x'}\|^2) \\[5pt]
&= \sum_{j=0}^\infty \frac{(\mathbf{x}^\top \mathbf{x'})^j}{j!} \exp\left(-\frac{1}{2}\|\mathbf{x}\|^2\right) \exp\left(-\frac{1}{2}\|\mathbf{x'}\|^2\right)\\[5pt]
&= \sum_{j=0}^\infty \quad \sum_{n_1+n_2+\dots +n_k=j}
\exp\left(-\frac{1}{2}\|\mathbf{x}\|^2\right)
\frac{x_1^{n_1}\cdots x_k^{n_k} }{\sqrt{n_1! \cdots n_k! }}
\exp\left(-\frac{1}{2}\|\mathbf{x'}\|^2\right)
\frac{{x'}_1^{n_1}\cdots {x'}_k^{n_k} }{\sqrt{n_1! \cdots n_k! }} \\[5pt]
&=\langle \varphi(\mathbf{x}), \varphi(\mathbf{x'}) \rangle
\end{alignat}
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