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Could someone tell me what I am missing here?

I want to show that for linear maps $L_1, L_2$, $v\in \ker(L_1)+\ker(L_2) \implies (L_1+L_2)v = 0$.

$v\in \ker(L_1)+\ker(L_2)$

$\implies \exists a,b\in V : L_1(a)=0=L_2(b)\wedge v=a+b$

$\implies (L_1+L_2)v = L_1(a)+L_2(a)+L_1(b)+L_2(b)$

$\implies (L_1+L_2)v = L_2(a)+L_1(b)$...

But then...? Thanks.

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