Proof that $a + 1 = 1$ (Boolean algebra)

Question

The proof works like this:

$$a + 1 = (a + 1) * 1 = (a + 1) * (a + \lnot a) = a + (1 * \lnot a) = a + \lnot a = 1$$

It only uses the axioms of Boolean algebra. I understand every step besides the third one. He uses the distributive law here but I simply don't see how.

Answer 1

There are two distributive laws in Boolean algebra (which are De Morgan duals of each other). The one in use here is $$(x+y)(x+z)=x+yz $$ with $x=a$, $y=1$, $z=\neg a$.

Answer 2

by using this property $a+a'=1 $

$a+1=a+(a+a')=a+a'=1$