On the other hand, however, Sperner's lemma which is used to prove Brouwer's theorem
can be constructively proved.
By applying Brouwer's Theorem
we can immediately speculate about the existence of a point x such that k(x) = x: (2)
0]) take on sets of values, as in the Kukutani fixed point theorem, which generalizes Brouwer's theorem
to a point-to-set mapping.
Such a function cannot be continuous, and so this instance of choice contradicts Brouwer's theorem
that every real-valued function is continuous.