Proof with induction even number of letter

I have to proof that in a word $ w$ the number of the letter d is always even.

Let $ L \subsetneq \Sigma^*$ be a language over the alphabet $ \Sigma = \{a,b,c,d\}$ such that a word $ w$ is in $ L$ if and only if it is $ a$ or $ b$ or of the form $ w = ducvd$ where $ u$ and $ v$ are word of $ L$ .


$ dacad$ , $ ddacbdcad$ , $ dddbcbdcdbcbddcad$

I know there are three cases:

  1. $ |w| = 0$ the number of the letter d is zero and even
  2. $ |w| = 1$ and $ w = a$ or $ w = b$ same as number 1
  3. $ |w| = 5$ and $ w = ducvd$ the number of letter d is even

    I know how induction works, but I don’t know to write the base, hypothesis and step. I think it works with the length of $ w$ . Does somebody have a hint?

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