Union of Unordered Tuples

Theorem
Let $x_1, \dots, x_n, x_{n+1}, \dots, x_m$ be arbitrary.

Then
 * $\left\{ {x_1, \dots, x_n}\right\} \cup \left\{ {x_{n+1}, \dots, x_m}\right\} = \left\{ {x_1, \dots, x_n, x_{n+1}, \dots, x_m}\right\}$

Proof
Let $a$ be arbitrary.


 * $a \in \left\{ {x_1, \dots, x_n}\right\} \cup \left\{ {x_{n+1}, \dots, x_m}\right\}$


 * $a \in \left\{ {x_1, \dots, x_n}\right\}$ or $a \in \left\{ {x_{n+1}, \dots, x_m}\right\}$ by definition of union


 * $a = x_1 \lor \dots \lor a = x_n$ or $a = x_{n+1} \lor \dots \lor a = x_m$ by definition of unordered tuple


 * $a = x_1 \lor \dots \lor a = x_n \lor a = x_{n+1} \lor \dots \lor a = x_m$


 * $a \in \left\{ {x_1, \dots, x_n, x_{n+1}, \dots, x_m}\right\}$ by definition of unordered tuple

Thus result fallows by definition of set equality.