I have quoted the solution you have given.

you have defined $\displaystyle $A_0 = A \setminus Gx$.

I agree with the fact that $\displaystyle $A_0 \cap Gx=\emptyset$, because this follows from the definition of $\displaystyle $A_0$.

But i don't agree with $\displaystyle \sigma($A_0) \cap $A_0=\emptyset$. According to me this is only true if G acts transitively on A. i say this because if G does not act transitively on A then $\displaystyle $A_0$ is non-empty( as in this case $\displaystyle Gx \subset A$), and there must be some element $\displaystyle a \in $A_0$ such that $\displaystyle \sigma(a) \notin Gx \text{ } \forall \text{ } \sigma \in G $.

May be i am just being a 'block'head here but i am only a newbie at group theory so please help this poor guy.