Existence and Uniqueness of the Block Matrix with Two Row Blocks and Two Column Blocks

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $m,n\in\mathbb{N}$, and for $p\in\mathbb{N}$ let $[p]$ be as in Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices §indices. Let $A$, $B$, $C$ and $D$ be real matrices of sizes $m\times m$, $m\times n$, $n\times m$ and $n\times n$ respectively, with entries $A_{kl}$, $B_{kj}$, $C_{il}$ and $D_{ij}$. By Splitting the Segment [m+n] into [m] and a Shifted Copy of [n] §sum and Splitting the Segment [m+n] into [m] and a Shifted Copy of [n] §shift, $m+n\in\mathbb{N}$, $[m]\subseteq[m+n]$ and $m+i\in[m+n]$ for $i\in[n]$, so all indices below lie in $[m+n]$.

¶ There is exactly one $M\in\mathcal{M}_{m+n}(\mathbb{R})$ such that, for all $k,l\in[m]$ and $i,j\in[n]$,

$$M_{kl}=A_{kl},\qquad M_{k,\,m+j}=B_{kj},\qquad M_{m+i,\,l}=C_{il},\qquad M_{m+i,\,m+j}=D_{ij}.$$