Linear mapping/Eigenspace for different eigenvalues/Zero/Fact

Let ${}K$ be a field, ${}V$ a ${}K$-vector space and

$\varphi \colon V\longrightarrow V$

a linear mapping. Let ${}\lambda _{1}\neq \lambda _{2}$ be elements in ${}K$.

Then

${}\operatorname {Eig} _{\lambda _{1}}{\left(\varphi \right)}\cap \operatorname {Eig} _{\lambda _{2}}{\left(\varphi \right)}=0\,.$

Write a proof