For the sake of simplicity the following notation $a_k := a[k]$ is assumed for time sequences.
A completely general discrete-time (DT) non-linear(NL) time-invariant (TI) dynamical system can be described with a system of vector difference equations, where $x[k]$ is the system state, $u[k]$ is the system input and $y[k]$ is the system output: $$\begin{cases} x[k+1] = f(x[k],u[k])\\ y[k] = g(x[k], u[k]) \end{cases} \quad\forall k \in \mathbb{Z} \ge0$$ $f(\cdot)$ and $g(\cdot)$ are general nonlinear functions.
The first equation is very close to a general autonomous non linear differential equation with $\dot x(t)=f(x(t),u(t))$ where $u(t)$ is the source term.
Now, the same dynamical DT NL TI system could be described also through an input output relationship, involving more among their "samples" (current and past): $$y[k] = h(y[k-1],y[k-2],...,y[k-n],u[k],u[k-1],...,u[k-m]), \quad\forall k \in \mathbb{Z} \ge 0 \quad$$ where again, $h$ is a general non linear function, and $n,m$ are integer positive values.
How can one prove that this input-output relationship form is equivalent to the former state-space representation, at least when $x[0]=0$ (without any care about a possible relationship between $f,g$ and $h$)?