Semantics of E and A operators in CTL* I'm referring to a book where E and A were defined in text as follows


  The formula $A \phi$ states that all the executions out of the current state satisfy property $\phi$ whereas $E\phi$ states that from the current state, there exists and execution satisfying $\phi$


And formally, they were defined as


  $\sigma, i \models E\phi $ iff there exists as $\sigma'$ such that $\sigma(0)\ldots\sigma(i) = \sigma'(0)\ldots\sigma'(i)$ and $\sigma', i \models phi$
  $\sigma, i \models A\phi $ iff for all $\sigma'$ such that $\sigma(0)\ldots\sigma(i) = \sigma'(0)\ldots\sigma'(i)$, we have $\sigma', i \models phi$


$\sigma(i)$ was previously defined to be the ith state of an execution $\sigma$. 

Here's where my confusion lies. Doesn't $\sigma(0)\ldots\sigma(i) = \sigma'(0)\ldots\sigma'(i)$ imply that the executions $\sigma$ and $\sigma'$ have both the same start and end state, therefore $\sigma', i \models \phi$ is same as $\sigma, i \models \phi$? That seems to be wrong.
I believe it means that $\sigma,\sigma$ should share the same prefix up to position $i$.
Then the quantification ranges over all possible extensions of this prefix from state $\sigma(i) = \sigma(i)'$.