We need to solve the equation $y' = 0$ (for the class of generalized functions where $y = (y,\varphi)$ and $(y',\varphi) = (y,-\varphi')$.
The textbook I am currently reading uses the following statements to prove that the solution to this equation is $y = C (C = const)$:
1) $\varphi_0 {(x)} = \varphi_1' (x)$ iff $ \int\limits_{-\infty}^{+\infty}{\varphi_0 (x)dx = 0}$
2) Given (1), we assume: $\varphi_1 (x) = \int\limits_{-\infty}^{x}{\varphi_0 (t)dt}$
3) Now let $\varphi_1 (x)$ be a fixed function with the property: $\int\limits_{-\infty}^{+\infty}\varphi_1(x)dx = 1$
4) For every $\varphi(x)$ we can write the following equality: $\varphi(x) = \varphi_1(x)\int\limits_{-\infty}^{+\infty}\varphi(x)dx + \varphi_0 (x)$ where $\varphi_0(x)$, obviously, satisfies (1).
What I can't understand is the fourth equation. How is it derived? The equality looks as if integration by parts was used, but I can't get the same result.
What am I missing?