If , then M eventually enters a final state. At this point, the derivation in G can use rule 8 to reproduce the original string w from the first component of the representation of every nonterminal in the resulting string. All the q's can be erased by using q, as many times as required. Therefore Sw and so wL(G).

Conversely, if wL(G) there is a derivation of w in G. Proceeding in exactly in opposite direction as discussed above, we discover that for some ,and . Hence , completing the proof.