  In 2012 Gamayun, Iorgov, Lisovyy conjectured an explicit expression for the
Painlev\'e VI $\tau$~function in terms of the Liouville conformal blocks with
central charge $c=1$. We prove that proposed expression satisfies Painlev\'e VI
$\tau$~function bilinear equations (and therefore prove the conjecture).
  The proof reduces to the proof of bilinear relations on conformal blocks.
These relations were studied using the embedding of a direct sum of two
Virasoro algebras into a sum of Majorana fermion and Super Virasoro algebra. In
the framework of the AGT correspondence the bilinear equations on the conformal
blocks can be interpreted in terms of instanton counting on the minimal
resolution of $\mathbb{C}^2/\mathbb{Z}_2$ (similarly to Nakajima-Yoshioka
blow-up equations).
