Suppose $T$ is an $L$-theory that has quantifier-elimination. Let $M$ be a $T$-model and $\Delta$ the set of all quantifier-free $L$-sentences that are true in $M$. Define $T'$:=$T$$\cup$$\Delta$. Now prove that $T'$ is a complete theory.
Is the following arguing correct?
Any two models $M,N$ ($M$ $\subset$ $N$) of $T'$ satisfy the same sentences ($T$ admits quantifier elimination and every sentence in $\Delta$ is a quantifier free sentence) and in this way $T'$ is a complete theory?