化簡 $$$\overline{\left(\overline{A} + B\right) \cdot \left(\overline{B} + C\right)}$$$

將德摩根定律 $$$\overline{x \cdot y} = \overline{x} + \overline{y}$$$ 應用於 $$$x = \overline{A} + B$$$ 與 $$$y = \overline{B} + C$$$：

$${\color{red}\left(\overline{\left(\overline{A} + B\right) \cdot \left(\overline{B} + C\right)}\right)} = {\color{red}\left(\overline{\overline{A} + B} + \overline{\overline{B} + C}\right)}$$

將德摩根定律 $$$\overline{x + y} = \overline{x} \cdot \overline{y}$$$ 應用於 $$$x = \overline{A}$$$ 與 $$$y = B$$$：

$${\color{red}\left(\overline{\overline{A} + B}\right)} + \overline{\overline{B} + C} = {\color{red}\left(\overline{\overline{A}} \cdot \overline{B}\right)} + \overline{\overline{B} + C}$$

以 $$$x = A$$$ 套用雙重否定（對合）律 $$$\overline{\overline{x}} = x$$$：

$$\left({\color{red}\left(\overline{\overline{A}}\right)} \cdot \overline{B}\right) + \overline{\overline{B} + C} = \left({\color{red}\left(A\right)} \cdot \overline{B}\right) + \overline{\overline{B} + C}$$

將德摩根定律 $$$\overline{x + y} = \overline{x} \cdot \overline{y}$$$ 應用於 $$$x = \overline{B}$$$ 與 $$$y = C$$$：

$$\left(A \cdot \overline{B}\right) + {\color{red}\left(\overline{\overline{B} + C}\right)} = \left(A \cdot \overline{B}\right) + {\color{red}\left(\overline{\overline{B}} \cdot \overline{C}\right)}$$

以 $$$x = B$$$ 套用雙重否定（對合）律 $$$\overline{\overline{x}} = x$$$：

$$\left(A \cdot \overline{B}\right) + \left({\color{red}\left(\overline{\overline{B}}\right)} \cdot \overline{C}\right) = \left(A \cdot \overline{B}\right) + \left({\color{red}\left(B\right)} \cdot \overline{C}\right)$$