Prove that
$[¯ a + ¯ b ¯ b + ¯ c ¯ c + ¯ a] = 2 [¯ a ¯ b ¯ c]$ .

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Solution

$[\to a + \to b \to b + \to c \to c + \to a] = (\to a + \to b) \cdot [(\to b + \to c) \times (\to c + \to a)]$
$= (\to a + \to b) \cdot [\to b \times (\to c + \to a) + \to c \times (\to c + \to a)]$
As $\to c | | \to c$
$\Rightarrow \to c \times (\to c \times \to a) = 0$
$= (\to a + \to b) \cdot [(\to b \times \to c) + (\to b \times \to a) + (\to c \times \to a)]$
$= \to a \cdot (\to b \times \to c] + \to a \cdot (\to b \times \to a) + \to a \cdot (\to c \times \to a) + \to b \cdot (\to b \times \to c) + \to b \cdot (\to b \times \to a) + \to b \cdot (\to c \times \to a)$
$= [\to a \to b \to c] + [\to a \to b \to c]$
$= 2 [\to a \to b \to c]$ .

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