If (kappa_{1}) and (kappa_{2}) are kernels on (mathscr{X}) and (mathscr{Y}), then (kappa_{+},left((boldsymbol{x}, boldsymbol{y}),left(boldsymbol{x}^{prime}, boldsymbol{y}^{prime} ight) ight):=kappa_{1}left(boldsymbol{x}, boldsymbol{x}^{prime}

If \(\kappa_{1}\) and \(\kappa_{2}\) are kernels on \(\mathscr{X}\) and \(\mathscr{Y}\), then \(\kappa_{+},\left((\boldsymbol{x}, \boldsymbol{y}),\left(\boldsymbol{x}^{\prime}, \boldsymbol{y}^{\prime}\right)\right):=\kappa_{1}\left(\boldsymbol{x}, \boldsymbol{x}^{\prime}\right)+\kappa_{2}\left(\boldsymbol{y}, \boldsymbol{y}^{\prime}\right)\) and \(\kappa_{\mathrm{x}}\left((\boldsymbol{x}, \boldsymbol{y}),\left(\boldsymbol{x}^{\prime}, \boldsymbol{y}^{\prime}\right):=\kappa_{1}\left(\boldsymbol{x}, \boldsymbol{x}^{\prime}\right) k_{2},\left(\boldsymbol{y}, \boldsymbol{y}^{\prime}\right)\right.\) are kernels on the Cartesian product \(\mathscr{X} \times \mathscr{Y}\). Prove this.