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 .