Let be the representative of the string in G. Consider the ID in G. Consider the ID of M for .

If (q, a_j)=(P, X, R) is a move of M, then using rule b, we find that

which is a correct representation of the next ID and it remembers the symbol in the first component of the nonterminal and modifies it to X in the second component.

If on the other hand, is move of M, then similarly, it is easy to see that using rule 7 we find a correct representation of the next ID of M.

Hence, at every step, using rule 6 and 7, the grammar G correctly simulates the computations of M.