How can suppose, let, and consider be used in proof writing?

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Homework Statement


This isn't a homework problem, but I just had a general question about the language of writing proofs. I often see words like "suppose", "let", and "consider" in proofs. Could someone explain to me if there are rules to applying these? It doesn't seem like they are just randomly thrown into start off a proof.


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I don't know of any formal rules. English has a lot of flexibility. All three of those words could be used to accomplish the same thing:

Suppose A is a right triangle, then the Pythagorean theorem...

Let A be a right triangle, then the Pythagorean theorem...

Consider a right triangle A. The Pythagorean theorem...

They all three serve to establish the hypothesis that A is a right triangle.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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