You can invoke
to show that
unique.
Defining now , you will have proven what is required.
let F be a closed subspace of a hilbert space H. Let P:H->F be defined by ||x-P(x)||<_ ||x-y|| for every y in F. In other words P(x) is orthogonal projection of x on F.
Show that orthogonal projection of x on the orthogonal complement of F is = to the identity operator - orthogonal projection of x on F.
Thanks