I want to prove $\vec A \times (\vec B \times \vec C)=-(\vec A \cdot \vec C)\vec B +(\vec A \cdot \vec B)\vec C $,here are my two proofs below
$\vec A \times (\vec B \times \vec C)=m\vec B+n\vec C$
$\vec A \cdot [\vec A \times (\vec B \times \vec C) ]=0=\vec A \cdot (m\vec B+n\vec C)$
$\vec A \cdot (m\vec B+n\vec C)=0=m(\vec A\cdot \vec B)+n(\vec A\cdot \vec C)$ , that is ,$m(\vec A\cdot \vec B)=-n(\vec A\cdot \vec C)$
Now here is the problem
1. If i assume $m=-(\vec A\cdot \vec C)$ and $n=(\vec A\cdot \vec B)$,it will become $-(\vec A\cdot \vec C)(\vec A\cdot \vec B)=-(\vec A\cdot \vec B)(\vec A\cdot \vec C)$,and it is equivalent
so i can say $\vec A \times (\vec B \times \vec C)=-(\vec A \cdot \vec C)\vec B +(\vec A \cdot \vec B)\vec C $
2. If i assume $m=(\vec A\cdot \vec C)$ and $n=-(\vec A\cdot \vec B)$,it will become $(\vec A\cdot \vec C)(\vec A\cdot \vec B)=(\vec A\cdot \vec B)(\vec A\cdot \vec C)$,and it is equivalent too
so i can say $\vec A \times (\vec B \times \vec C)=(\vec A \cdot \vec C)\vec B -(\vec A \cdot \vec B)\vec C $
However,$\vec A \times (\vec B \times \vec C)=-(\vec A \cdot \vec C)\vec B +(\vec A \cdot \vec B)\vec C \neq (\vec A \cdot \vec C)\vec B -(\vec A \cdot \vec B)\vec C $,so i want to ask where am i wrong?