  We investigate Gale's important paper published in 1960. This paper contains
an example of a candidate of the demand function that satisfies the weak axiom
of revealed preference and that is doubtful that it is a demand function of
some weak order. We examine this paper and first scrutinize what Gale proved.
Then we identify a gap in Gale's proof and show that he failed to show that
this candidate of the demand function is not a demand function. Next, we
present three complete proofs of Gale's claim. First, we construct a proof that
was constructible in 1960 by a fact that Gale himself demonstrated. Second, we
construct a modern and simple proof using Shephard's lemma. Third, we construct
a proof that follows the direction that Gale originally conceived. Our
conclusion is as follows: although, in 1960, Gale was not able to prove that
the candidate of the demand function that he constructed is not a demand
function, he substantially proved it, and therefore it is fair to say that the
credit for finding a candidate of the demand function that satisfies the weak
axiom but is not a demand function is attributed to Gale.
