In a proof for the fact that every distribution has an antiderivative (in Introduction to Hilbert space with application(Debnath,Mikusinski)) they use the fact that for any $$ \phi_0 \in D(R) $$ s.t $$ \int_{-\infty}^{\infty}\phi_0(x)dx = 1$$ we can represent every test function $$ \phi \in D(R), \space \space \phi = K\phi_0 + \phi_1$$ where $$ K = \int_{-\infty}^{\infty}\phi(x)dx, \space \int_{-\infty}^{\infty}\phi_1(x)dx = 0$$ now I'm stuck on an exercise that wants me to prove the representation is possible for every phi aswell as the fact that the represention is unique, I've tried to reach some equality between the two when I apply a distribution but my understanding of distributions and test function is very limited aswell as my intuition for these kinds of problems.
I'm having trouble getting started in the right direction so any help is appreciated.
Fix $\phi_0 \in \mathcal{D}(\mathbb R)$ such that $\int_{-\infty}^{\infty} \phi_0(x) \, dx = 1$.
For an arbitrary $\phi \in \mathcal{D}(\mathbb R)$ let $K = \int_{-\infty}^{\infty} \phi(x) \, dx$ and $\phi_1 = \phi - K\phi_0.$
Show that $\phi_1 \in \mathcal{D}(\mathbb R)$. This shows that "the representation is possible for every phi".
Assume that there is another represention $\phi = L\phi_0 + \phi_2,$ where $L = \int_{-\infty}^{\infty} \phi(x) \, dx$ and $\int_{-\infty}^{\infty} \phi_2(x) \, dx = 0.$ Show that $K=L$ and $\phi_1=\phi_2.$ This proves uniqueness.