Eigenspaces/Direct sum/Dimension/Fact/Proof

The directness means

${\displaystyle {}\operatorname {Eig} _{\lambda _{k}}{\left(\varphi \right)}\cap {\left(\sum _{i\neq k}\operatorname {Eig} _{\lambda _{i}}{\left(\varphi \right)}\right)}=0\,}$

for every ${\displaystyle {}k}$. Suppose that this is not the case. Then we had vectors ${\displaystyle {}v_{i}\in \operatorname {Eig} _{\lambda _{i}}{\left(\varphi \right)}}$ fulfilling

${\displaystyle {}0\neq v_{k}=\sum _{i\neq k}v_{i}\,.}$

But this contradicts fact.