Yes, basic math. c^2 = a^2 + b^2 and area = ab/2. So, picking a for example, a = sqrt(c^2 - b^2) and thus A = b*sqrt(c^2-b^2)/2. Area is a function of the hypotenuse with the caveat that b can take any value as long as b < c. This is why the iPhone X has a mathematically larger surface area. You can't beat math.
Wow.
You do realize you need to compare A1 to A2. Which means b for each triangle actually does matter. Your proof doesn't actually prove what you're saying. It does show that you know how to isolate variables in equations, though.