first of all, realize that is an equality of two sets.
so suppose . then for some .
so and because is orthogonal, , that is
so , so .
now suppose we have , so that .
since is an orthogonal automorphism, it is bijective, and its inverse is also orthogonal.
so . but .
this means that so ,
which shows that , thus the two sets are equal.
the second equality, is an equality of functions, so we must show that the two functions have the same value at every point of .
so let . since is surjective, we can write for some ,
where . so then we have:
for all .
this shows the two functions are indeed equal on all of .