To prove the equation
, we will manipulate the given expression using Boolean algebra laws and logic operations.
Starting with the left-hand side of the equation:

We can expand the term
using the XOR operation:

Applying De Morgan's law to the expression inside the complement:

Distributing the complement over the product:

Using De Morgan's law again:

Now, let's simplify the right-hand side of the equation:

Factoring out A from the second term and B from the third term:

Applying the distributive law:

Using the absorption law, which states that
:

Comparing the right-hand side expression with the expression (1) derived from the left-hand side, we can see that they are equal.
Therefore, we have proven that
.

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