I've tryed to prove it, but i didn't succeed. By the way i think I've found a more directly way to show that ##H=[g_1 H]## if ## g_1 \in H##, but I'm stuck:
Let's (## \forall g_1 , g_2 \in H##) build two left cosets, ##[g_1 H]## and ##[g_2 H]##, now, suppose that ##\exists \tilde{g}## so that...