$3\mathrm{.}12$ Prove algebraically that the following equ

Prove algebraically that the following equation

$A^{\text{'}}C{D}^{\text{'}}E+A^{\text{'}}{B}^{\text{'}}{D}^{\text{'}}+$ABCE$+$ABD$=A^{\text{'}}{B}^{\text{'}}{D}^{\text{'}}+$ABD$+BC{D}^{\text{'}}E$

$3\mathrm{.}13$ Simplify each of the following expressions:

$($a$)KLM{N}^{\text{'}}+K^{\text{'}}{L}^{\text{'}}MN+M{N}^{\text{'}}$

$($b$)K{L}^{\text{'}}{M}^{\text{'}}+M{N}^{\text{'}}+L{M}^{\text{'}}{N}^{\text{'}}$

$($c$)(K+L^{\text{'}})(K^{\text{'}}+L^{\text{'}}+N)(L^{\text{'}}+M+N^{\text{'}})$

$($ci$)(K^{\text{'}}+L+M^{\text{'}}+N)(K^{\text{'}}+M^{\text{'}}+N+R)(K^{\text{'}}+M^{\text{'}}+N+R^{\text{'}})KM$