Proof that the construction on page 46 is correct.

Proof that the machine M recognizes the language A1 union A2.  

Let w = w1, w2, ..., wn be a string over sigma.

Let ro, r1, ..., rn be the sequence of states that 
M enters to process w, where one(ri) is the state 
from M1 labeling ri and two(ri) is the state 
from M2 labeling ri.

------
Lemma:  Let S be a string, and let ri be the state 
that M enters after consuming S.  Show that M1 
is in state one(ri) after consuming S and M2 is in state 
two(ri) after consuming S.

Proof by induction: Base case: Let S be the empty string.  
   M is in (q1,q2).  M1 is in q1, and M2 is in q2, since 
   those are the start states.

   Let S be a string of length m, and rm be the state that M 
   enters after consuming S.  By the induction hypothesis, 
   M1 is in state one(rm) and M2 is in state two(rm) after 
   consuming S. Let s be the next input to consume.  By the 
   construction procedure, M enters the state 
   (delta-1(one(rm),s), delta-2(two(rm),s)) after consuming S.  
------  

Since wn is a final state of M iff one(wn) is a final 
state of M1 or two(wn) is a final state of M2 
(by the construction procedure), M accepts w
iff M1 or M2 accepts w.