Formal Language $\gamma$

Question

Suppose a formal language $\gamma$ is defined with $\Sigma=\left\{M,I,U\right\}$ and the following rules:

let $x,y \in \gamma$

1. $MI$ is in the language
2. If $xI$ is in the language then so is $xIU$
3. If $xIIIy$ is in the language then so is xUy
4. If $Mx$ is in the language is then so is $Mxx$
5. If $xUy$ is in the language then so is $xy$

The following theorem has been stated in our course notes: If $xIII$ is in $\gamma$ so is $x$

I have tried playing around with the rules of the language but I haven't been able to get the desired result, could someone show me how?

Answer 1

HINT

$y$ can be the empty string.