Set-up. Consider two linear combinations $$ y^{(1)}=\sum_{i=1}^{N-1}a_{i}x_{i}+a_{N}x_{N} \hspace{2cm} (1)\\ y^{(2)}=\sum_{i=1}^{N-1}a_{i}x_{i}+a_{N}x_{N} \hspace{2cm} (2) $$ where $a_{i} \in \mathbb{R}_{\geq 0}$, each $x_{1,...,N-1} $ is independently distributed with continuous density function $f_{i}(x)$, $z_{N}$ is independently distributed with density $h^{(1)}(x_{N})$ in (1) and $x_{N}$ is independently distributed with density $h^{(2)}(x_{N})$ in (2). Moreover, $h^{(1)}(x_{N})$ first order stochastically dominates $h^{(2)}(x_{N})$; i.e. $H^{(1)}(x_{N})\leq H^{(2)}(x_{N}) \ \forall x_{N}$.
Question. Is there a way to show that if $H^{(1)}(x_{N})\leq H^{(2)}(x_{N})$, then the density function which governs $y^{(1)}$ stochastically dominates the density function of $y^{(2)}$?
The proof of this can be found in theorem 5 in
Hadar, J., & Russell, W. R. (1971). Stochastic dominance and diversification. Journal of Economic Theory, 3(3), 288-305.