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$$\text{Output of upper NOR gate} = \overline{A + A} = \bar{A}$$
$$\text{Output of lower NOR gate} = \overline{B + B} = \bar{B}$$
Final output from the rightmost NOR gate: $$Y = \overline{\bar{A} + \bar{B}}$$
Applying De Morgan's law: $$Y = \overline{\overline{A}} \cdot \overline{\overline{B}} = A \cdot B$$
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