  We give a proof of the Nekrasov-type formula proposed by one of the authors
for the Seiberg-Witten prepotential for the E-string theory on R^4 x T^2. We
take the thermodynamic limit of the Nekrasov-type formula following the example
of Nekrasov-Okounkov and reproduce the Seiberg-Witten description of the
prepotential. The Seiberg-Witten curve obtained directly from the Nekrasov-type
formula is of genus greater than one. We find that this curve is transformed
into the known elliptic curve by a simple map. We consider the cases in which
the low energy theory has E_8, E_7+A_1 or E_6+A_2 as a global symmetry.
