Since $T(y)$ is globally insufficient, there exist a $T_1$ and sets of positive measure $Y_{11}$ and $Y_{12}$ which are disjoint and subsets of $Y_1=(y \vert T(y)=T_1)$ such that: $$ \frac{g_a(Y_{11},a)}{g(Y_{11},a)} \not= \frac{g_a(Y_{12},a)}{g(Y_{12},a)} \qquad (1) $$
solve the system: $$ ds_{11}g(Y_{11},a)+ds_{12}g(Y_{12},a)=-ds_{2}g(Y_{2},a) \qquad (2) $$ $$ ds_{11}g_a(Y_{11},a)+ds_{12}g_a(Y_{12},a) =\frac{u^\prime_2 }{u^\prime_1} ds_{2}g_a(Y_{2},a) \qquad (3) $$
The entire proof to which my question refers
The system (2) and (3) has a solution because (1) implies that the determinant is nonzero. Can someone explain why (1) implies that the system has a solution?
Thanks in advance and if something is unclear about the question please let me know :)
according to (1) $ g_{a}(Y_{11},a)g(Y_{12},a)≠g_{a}(Y_{12},a)g(Y_{11},a) $ now try to get the determinant for your system. It can only be zero, if $ g_{a}(Y_{11},a)g(Y_{12},a) - g_{a}(Y_{12},a)g(Y_{11},a) = 0 $, but it's not according to the forementioned info.