Textbook Problem

In 1-3 assume that B is a Boolean algebra with operations = and ‘ . Give the reasons needed to fill in the blanks in the proofs using only the axioms for a Boolean algebra.

1. Idempotent law for’ : For every a in B, ???
Proof: Let a be any element of B. Then
$\begin{array}{c}a=a.1(a)\\ =a.\left(a+\bar{a}\right)(b)\\ =\left(a\cdot a\right)+\left(a\cdot \bar{a}\right)(c)\\ \text{=}\left(a\cdot a\right)+0\text{(d)}\\ =a.a(e)\end{array}$