On the Decomposition of the Convolution of Gaussian and Poissonian Laws

The paper contains a detailed proof for the following theorem: the convolution of a Gaussian and a Poissonian law can be decomposed only into similar convolutions. More precisely, let $X=X_1+X_2$, where $X_1$ is a Gaussian component and $X_2$ a Poissonian component independent of $X_1$. If there is some other decomposition: $X=Y_1+Y_2,Y_1$ being independent of $Y_2$, then
$$Y_1=Y_{11}+Y_{12},\\Y_2=Y_{21}+Y_{22}.$$
where $Y_{11},Y_{21}$ are Gaussian and $Y_{12},Y_{22}$ Poissonian, all mutually independent, and
$$\mathbf D\left(Y_{11}\right)+\mathbf D\left(Y_{21}\right)=\mathbf D\left(X_1\right),\\\mathbf D\left(Y_{12}\right)+\mathbf D\left(Y_{22}\right)=\mathbf D\left(X_2\right).\\$$
Thus, H. Cramer's theorem on decomposing the normal law, and D. A. Raykov's theorem on decomposing the Poissonian law are special cases of this theorem.