Simple property of DFAs Do you know if this simple property of a DFA is stated as a property (or theorem) in some Automata theory book (possibly with a particular name)?

Property: Given a DFA $A = \{ Q, \Sigma, \delta, q_0, F \}$, if $w = uv$ and the state of $A$ on input $w$, after scanning $u$ is $q_i$ (when the head is at the beginning of subword $v$); and $A' = \{ Q, \Sigma, \delta, q_i, F \}$ then $w \in L(A)$ if and only if $v \in L(A')$
Actually I think I have spotted this property in a book. It uses the "extended" transition relation $\delta^*$, i.e., for strings rather than single letters. 


  Other properties you would expect $\delta^*$ to satisfy can be derived from
  our definition. For example, a natural generalization of the recursive
  statement in the definition is the formula 
  $$\delta^*(q,xy) = \delta^*(\delta^*(q,x),y))$$ 
  which is true for every state $q$ and every two strings $x$ and $y$ in $\Sigma^*$.
  The proof is by
  structural induction on $y$ and is similar to the proof [...] in Example 1.27.
  
  John C. Martin, Introduction to Languages and the Theory of Computation, McGraw-Hill, 4th edition, c2011. Page 54.


I hope you agree with me that this matches the "machine" terminology you use in your question.