Consider the following DFA:
It is quite clear that the language of this FDA is all the words that don't have the word $aa$ as a subword.
My question is: How can I formally prove that this is the language of this FDA ?
My efforts: I tried to determine $L(q_0)$ and $L(q_1)$ (that we denote as $L_0$ and $L_1$ accordingly) and prove that these are indeed what I determined using induction (this is the type of method used in the book I am studing from), I had some problems determining $L(q_0)$ and $L(q_1)$ and I am not sure if I should show equality 'straight out' or should I do two proofs showing $L(q_i)\subset L_i$ and $L_i\subset L(q_i)$.
Help is very much appreciated!