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So:

[tex]f = x_1 \overline{x_2} + \overline{x_1}x_2 [/tex]

If we let [tex]\uparrow[/tex] represent the NAND function. That means that: [tex]f = (x_1 \uparrow \overline{x_2}) \uparrow (\overline{x_1} \uparrow x_2) [/tex]

I follow everything up to that step. And then they attempt to decompose it by manipulating one of the terms.

[tex] (x_1 \uparrow \overline{x_2} ) = \overline{x_1 \overline{x_2}} = \overline{x_1 (\overline{x_1} + \overline{x_2})} = x_1 \uparrow (\overline{x_1} + \overline{x_2}) = x_1 \uparrow (x_1 \uparrow x_2)[/tex]

Can anyone please explain what exactly went on in that step? How did they go from the second equation to the third, and then the fourth? I understand the first and final steps, but that's it.

Any help would be greatly appreciated!