Let $(X,\Phi,d_X)$ and $(Y,\Pi,d_Y)$ be topological systems. Suppose $(X,\Phi,d_X)$ and $(Y,\Pi,d_Y)$ are distal, that is for any $x_1,x_2\in X$ with $x_1\not=x_2$, one has $$\inf\limits_{t\in\mathbb{R}}d_X(\Phi_t(x_1),\Phi_{t}(x_2))>0.$$ Similarly for $(Y,\Pi,d_Y)$.
Assume $p$ is a flow homomorphism, that is $p:X\rightarrow Y$ is a continuous map which preserves flows, that is $$\Pi_{t}\circ p(x)=p\circ\Phi_{t}(x).$$ If $p$ exists, we call $(Y,\Pi,d_Y)$ is a factor of $(X,\Phi,d_X)$.
My question is if $(Y,\Pi,d_Y)$ is a factor of $(X,\Phi,d_X)$ and $(Y,\Pi,d_Y)$ is distal, how to prove $(X,\Phi,d_X)$ is distal?