  We start from an interpretation of the $BC_2$-symmetric "Type I" (elliptic
Dixon) elliptic hypergeometric integral evaluation as a formula for a
Casoratian of the elliptic hypergeometric equation, and give an extension to
higher-dimensional integrals and higher-order hypergeometric functions. This
allows us to prove the corresponding elliptic beta integral and transformation
formula in a new way, by proving both sides satisfy the same difference
equations, and that the difference equations satisfy a Galois-theoretical
condition that ensures uniqueness of simultaneous solution.
