What is the logical reasoning behind <PERSON>'s Theorem proof of unique solution?
Geeksforgeeks contains a proof of <PERSON>'s theorem, asserting that $R=QP^$ is the unique* solution to $R=Q+RP$. The proof is reproduces below.
My question is:
What is the logical reasoning to prove that any equation is the unique (only solution)? Particularly in this case, how can the procedure below logically lead to the proof that R=QP must be the unique (only solution)*?
Here is my understanding of the proof:
1. Recursively substitute $R$ in $R=Q+RP$ with $Q+RP$
2. Establish the recursive definition of $R$
3. Generalize the definition to $R=QP^*$
Disclaimer: The proof below is second part of a 2-parts proof from the original Geeksforgeeks's proof of <PERSON>’s Theorem. Originally, the proof starts out to prove $R=QP^$ is a solution to $R=Q+RP$. I omitted this part here because I want to focus on the second part of the proof, which is to prove uniqueness. However, to clear any confusions, please allow the proof below to be based on the assumption that $R=QP^$ is a solution to $R=Q+RP$ has been correctly proven prior and that $R=QP^$ can be used as a corollary for proof below, which try to prove $R=QP^$ is the unique solution to $R=Q+RP$, given P does not contain $\epsilon$**
Given that $P$ and $Q$ are two regular expressions over $\Sigma$, and $P$ does not contain $\epsilon$. 330
Start with: $$R = Q + RP$$
Now, replace $R$ by $R = Q + RP$: $$ \begin{align} R &= Q + (Q + RP)P \ &=Q + QP + RP^2 \end{align} $$
Again, replace $R$ by $R = Q + RP$: $$ \begin{align} R &= Q + QP + (Q+RP)P^2 \ &= Q + QP + QP^2 + RP^3 \ &= \cdots \ &= Q + QP + QP^2 + \cdots + QP^n + RP^{n+1} \end{align} $$
Now, replace $R$ by $R = QP^$ to get $$ R = Q + QP + QP^2 + \cdots + QP^n + QP^P^{n+1} $$
Taking $Q$ as a common factor, $$ \begin{align} R &= Q(\epsilon + P + P^2 + \cdots + P^n + P^P^{n+1}) \ &= QP^, \end{align} $$ as $\epsilon + P + P^2 + \cdots + P^n + P^* P^{n+1}$ represents the closure of $P$.
Thus, $R = QP^*$ is the unique solution to $R = Q + RP$.
Clarifications:
I asked this question with the assumption that this is a valid proof $R=QP^$ is the unique* solution to $R=Q+RP$ (a). There are two basis for my assumption:
(1) This is the most popular proof found on the Web, I have included multiple sources as samples below. Across different authors, the proof takes exactly the same form as the procedure demonstrated above. Therefore, through inductive reasoning, I believe that this proof and its similar forms is a valid proof of (a). Otherwise, all those authors somehow collectively give a false proof.
(2) The procedure is logically and mathematically valid at each and every step, including the generalization of P* given that P does not contain $\epsilon$ as demonstrated above. Therefore, through deductive reasoning and (1), I believe that this proof and its similar forms is a valid proof of (a) until otherwise disproved through valid and sound counterargument.
Of course, what missing here is the intuitive reasoning from which one can derive that this proof indeed validly proves (a) to be true. Moreover, I am very curious and interested in the general intuitive and logical reasoning to prove anything as a unique solution.
I am not asking for a new proof. If for logical and/or mathematical reason, you can prove that this proof is invalid and MUST be discarded. Please present your evidence/counterargument in your answer. 945