Exercises 12-15 provide an outline for a proof that the associative laws, which were included as axiom for a Bolllem algebre, can be derived from the other four axioms. The algebra, can be derived from the four axioms. The outline is from Introduction to Boolean Algebra by S. Givant and P. Halmos, Springer, 2009. In order to avoid unneeded parentheses, assume that takes precedence over+.

Test for equality law: For all elements a,b,and c ns a Boolean algebra.

   If   b ⋅ a = c ⋅ a   and   b ⋅ a ¯ = c . a ¯ = c . a ¯ ,   then   b = c .

Without using the associative law, derive this law from the other four laws in the axioms for a Boolean algebra plus the result of exercise 12.