# Proving Flow Property of Stochastic Differential Equation

I am trying to show that $$X_t^{s,x} = X_t^{r, X_r^{s,x}}$$ for $$0 \leq s \leq r \leq t$$, $$x \in \mathbb{R}^n$$ is a given initial condition for time $$s$$, for some SDE: $$\begin{equation*} d X(u)=b(X(u))d u+\sigma(X(u))d B(u). \end{equation*}$$

I have seen a couple of proofs that use the uniqueness of the solution to show this, but I feel like some of the proofs have some circular reasoning in it. I understand the general idea, that since the stochastic process $$X_t$$ is unique, then at some time $$t$$, if we can write two expressions for the process's random value, then the expressions must be equal. In Oksendal the proof is basically what I wrote above and is two lines: But I wanted a more 'formalized' version of this proof, but couldn't find one until I searched online and found this following one:

From the following link: https://www.math.tecnico.ulisboa.pt/~czaja/ISEM/10internetseminar200607.pdf, the proof of this is the following: I am wondering how to justify the step of 'replacing' $$Z_u$$ with $$X_u^{s,x}$$. Isn't this saying that $$Z_u = X_u^{s,x}$$, where $$Z_u = X_u^{r, X_r^{s,x}}$$, and so $$X_u^{s,x} = X_u^{r, X_r^{s,x}}$$. And if we are asserting that this is true, isn't this the same as what we are trying to prove: $$X_t^{s,x} = X_t^{r, X_r^{s,x}}$$. So I am not sure what I am missing here.

Any help would be greatly appreciated. Thanks!