(A + ~A) * B = 1 * B = B
A * (A + B) * (C + ~B) = (A + AB) * (C + ~B) = A * (C + ~B) = AC + A(~B)
A(~B) + BC + (~A)(~B) = (~B)(A + ~A) + BC = ~B + BC = ~B + C
A + (~A)B = A + B
A + ~( (~A) * B ) = A + (A + ~B) = A + ~B
~( (~A) * (~B) ) + ~A = (A + B) + ~A = 1