Dedekind modular law
Solution 1:
There is a mistake in your proof. When you state Dedekind's rule above, you assume $A, B, C$ are subgroups of a group. So, to get your second equality using Dedekind's rule you assume $B= HU$ is a group, but that's what you are trying to prove. (Recall, $UH = HU$ if and only if $UH$ is a group.)