Very strange of Mathematica!

The following two expressions are the same when checked numerically.

fun1[x_, y_] =    1/4 (Sqrt[(1 - y)/(       4 + y (-4 + (1 - 2 x)^2 y))] (2 - y + Sqrt[         4 + y (-4 + (1 - 2 x)^2 y)]) -       2 ((-1 + 2 x) y + Sqrt[         4 + y (-4 + (1 - 2 x)^2 y)]) Sqrt[-(((-1 + y) (-2 + y + Sqrt[           4 + y (-4 + (1 - 2 x)^2 y)])^2)/(4 -           4 y + ((-1 + 2 x) y + Sqrt[            4 + y (-4 + (1 - 2 x)^2 y)])^2)^2)]); fun2[x_, y_] = Sqrt[1 - y]/2; 

However, Mathematica is not able to reduce fun1[x,y] to fun2[x,y], with Simplify of FullSimplify. Is there any way of doing this simplification? Also note that $ 0\le x \le 1$ and $ 0\le y\le 1$ .