I'm wondering if there is a standard reference for the following straightforward facts about well-orderings. (They are quite easy to prove, I just need a handy/standard reference.)

Let $(S,<)$ be a well-ordered set. The set $T=\{(s_1,s_2,\ldots, s_k)\ : \ k\in \mathbb{N},s_i\in S\}$, of all ordered finite tuples, is well-ordered in the degree-lexicographical ordering $\prec$. In other words, given different elements $\vec{a}=(a_1,a_2,\ldots, a_m),\vec{b}=(b_1,b_2,\ldots, b_n)\in T$ then we have $\vec{a}\prec \vec{b}$ exactly when either (1) $m<n$ or (2) $m=n$, and if $i$ is the first index where $a_i\neq b_i$, then $a_i<b_i$.

Let $(S,<)$ be a well-ordered set. The set $U$ of all finite subsets of $S$ is well-ordered by the following relation: given different finite subsets $S_1,S_2\subset S$ we say $S_1\prec S_2$ if it happens that the maximal element of (the finite set) $(S_1\cup S_2)\setminus (S_1\cap S_2)$ belongs to $S_2$.