A flawed theorem about regular languages

I am struggling with this question for a very long time and just can't find the flaw. So I am given a false Theorem:

The language ${awwa \mid w \in {a,b}^* }% is regular. Well, that part is obvious, we can prove it using the pumping lemma. The question I am asked is to find the flaw in the "proof" for this theorem: Let$\qquad L_1 = \{ aw \mid w \in \{a,b\}^* \}\qquad L_1^R = \{ aw \mid w \in \{a,b\}^* \}$Let$L_2 = L_1 L_1^R = \{ awwa \mid w \in \{a,b\}^*\}$be even-length palindromes that begin and end with an$a$. Since$L_1$is regular, and the class of regular languages is closed under reversal and concatenation, we conclude$L_2$is also regular. Can you find the flaw? I could build a DFA for$L_1$and$L_1^R$, so I know they are regular. And regular languages are closed under reversal and concatenation. But, however,$L_2$is still not regular, so where is the mistake in the "proof"? - Try writing down some words from$L_1 \cdot L_1^R$-- you'll notice quickly what's wrong. (Note that you can use LaTeX here to typeset mathematics in a more readable way. See here for a short introduction.) – Raphael Feb 21 '14 at 11:50 2 Answers The concatenation of$L_1$and$L_1^{\mathrm{R}}$is not$\{awwa\mid w\in\{a,b\}^*\}$. - it's not? How come? – user14896 Feb 21 '14 at 1:42 @user14896 What is the definition of the concatenation of two languages? – David Richerby Feb 21 '14 at 1:43 Remember that for some languages$L_1$,$L_2$,$LL = \{uv \;\colon\; u \in L_1,\;v \in L_2\}$. Back to your question, is it the case that for every$au \in L_1$and$va \in L_1^R$that$auva$is an even length palindrome that begins and ends with$a\$?

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Ohhhh...I see! Thank you so much!!! –  user14896 Feb 21 '14 at 2:02