Here is a trick from one of the proofs in probability:
$$\iint \mathbf 1_{(x,x+a]}(y) \ \lambda(dx) \ \mathbb P(dy) = \iint \mathbf 1_{[y-a,y)}(x) \ \lambda(dx) \ \mathbb P(dy)$$
for $a>0$. So basically $$\mathbf 1_{(x,x+a]}(y) = \mathbf 1_{[y-a,y)}(x)$$
for any $x,y \in \mathbb R $.
I have never seen the trick before and the only proof I came up with was to consider four cases: whether $y \in (x,x+a], x \in [y-a,y)$, which is quite clumsy.
Is there a better proof? Is there a way to visualize it (in order to remember the way it goes)?
$$\mathbf 1_{(x,x+a]}(y)=1\iff x\lt y\leqslant x+a\iff y-a\leqslant x\lt y\iff \mathbf 1_{[y-a,y)}(x)=1$$