<s>[INST] Write down a mathematical theorem of your choice and prove it in great detail. [/INST] The Pythagorean theorem states that for a right triangle with catheti (legs) of length $a, b$ and a hypotenuse of length $c$:

$$
a^2 + b^2 = c^2 .
$$

Proof:
From four copies of a right triangle a square can be constructed by rotating and translating the copies in such a way that the hypotenuses are on the inside and the catheti on the outside. Such a square has a total area of $(a + b)^2$. Meanwhile the hypotenuses form a smaller square within the aforementioned square that has an area of $c^2$. Each copy of the right triangle has an area of $\frac{a b}{2}$. The area of the inner square must be equal to the area of the outer square minus the area of the four copies of the right triangle. This yields:

$$
(a + b)^2 - 4 \frac{a b}{2} = a^2 + 2 a b + b^2 - 2 a b = a^2 + b^2 \overset{!}{=} c^2.
$$

Q.E.D.</s>