Simplifique $$$\overline{\overline{A \cdot B} + \left(\overline{D} \cdot A\right)}$$$

Aplique o teorema $$$\overline{X + Y} = \overline{X} \cdot \overline{Y}$$$ de De Morgan com $$$X = \overline{A \cdot B}$$$ e $$$Y = \overline{D} \cdot A$$$:

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

Aplique a lei de dupla negação (involução) $$$\overline{\overline{X}} = X$$$ com $$$X = A \cdot B$$$:

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

Aplique o teorema $$$\overline{X \cdot Y} = \overline{X} + \overline{Y}$$$ de De Morgan com $$$X = \overline{D}$$$ e $$$Y = A$$$:

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

Aplique a lei de dupla negação (involução) $$$\overline{\overline{X}} = X$$$ com $$$X = D$$$:

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

Aplique a lei comutativa:

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

Aplique a lei comutativa:

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

Aplique a lei de redundância $$$X \cdot \left(\overline{X} + Y\right) = X \cdot Y$$$ com $$$X = A$$$ e $$$Y = D$$$:

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