The Determinant of a Triangular Matrix is the Product of its Diagonal Entries

Let $n$ be a natural number, let $[n]$ be the initial segment determined by $n$, ordered by the relations of Order on the Natural Numbers, and let $L$ be a real $n\times n$ matrix. Call $L$ lower triangular if $L_{il}=0$ whenever $i,l\in[n]$ and $i<l$, and upper triangular if $L_{il}=0$ whenever $i,l\in[n]$ and $l<i$. Determinants are read as in Row Properties of the Determinant, and the product below is the finite product over $[n]$.

Then the following hold.

1. (Lower triangular) If $L$ is lower triangular, then

2. (Upper triangular) If $L$ is upper triangular, then