I am stuck at a small thing in the proof of conjugacy of positive systems under a finite reflection group. I am using the notation and definitions used in the text by James E. Humphreys. I reproduce the proof below: Theorem:Let $\Delta$ be a positive system, contained in the positive system $\Pi$. If $\alpha \in \Delta$ then $s_\alpha(\Pi \backslash {\alpha})=(\Pi \backslash {\alpha})$
Proof: Let $\beta \in \Pi, \beta \neq \alpha$. Write $\beta =\sum_{\gamma \in \Delta}{c_\gamma \gamma}, c_\gamma \geq 0$. $c_\gamma > 0$ for some $\gamma \neq \alpha$ (I understand why). Now apply $s_\alpha$ to both sides: $s_\alpha \beta = \beta -c\alpha$ is a linear combination of $\Delta$ involving $\gamma$ with the same coefficient $c_\gamma$. Because all coefficients in such an expression have like sign, $s_\alpha \beta$ must be positive.
This is the part I do not understand. Why can't it happen that $c_\alpha \neq 0$ in the expression for $\beta$ and then $c_\alpha -c \lt 0$ in the expression for $s_\alpha \beta$? Then all coefficients in the expression will not have like sign.