I was working on showing that if $f \in L^1(\mathbb{R}^d)$ is a radial function, then so if its Fourier transform. I had a difficult time proving that, and sought help on MSE. Something had already asked a very similar question over here: Fourier transform of a radial function.
Now, one of the proofs is this one:
So suppose $f \in L^1(\mathbb{R}^d)$ and $f$ is radial. Fix an orthogonal transformation $T$. Then: $$\hat {f} (Tx) = \int_{\mathbb {R}^n} f(t) e^{-i\langle Tx,t \rangle}\,dt \overset{(1)}{=} \int_{\mathbb {R}^n} f(Ts) e^{-i\langle Tx,Ts \rangle}\,ds\overset{(2)}{=} \int_{\mathbb {R}^n} f(s) e^{-i\langle x,s \rangle}\,ds = \hat f (x).$$
I don't understand how the $(1)$ and $(2)$ appear. Can someone explicit better the line of thought given here?