  About twenty years ago we wrote a paper, "Boolean Functions whose Fourier
Transform is Concentrated on the First Two Levels", \cite{FKN}. In it we
offered several proofs of the statement that Boolean functions
$f(x_1,x_2,\dots,x_n)$, whose Fourier coefficients are concentrated on the
lowest two levels are close to a constant function or to a function of the form
$f=x_k$ or $f=1-x_k$. Returning to the paper lately, we noticed that the
presentation of the first proof is rather cumbersome, and includes several
typos. In this note we rewrite that proof, as a service to the public.
