Induction: For an arbitrary string x, if
, then 1x, 0x, x0, x1 are all in U. Extremal: Nothing is in U unless it is obtained using the above two clauses.
2. Basis:
Induction: For an arbitrary string x, if , then 1x, 0x, x0, x1 are all in U.
Extremal: Nothing is in U unless it is obtained using the above two clauses.
3. It is going to be proved for an arbitrary string x and an arbitrary state q by induction on string y.
Basis: If 
, then 
Also 
for FA.
Hence LHS = RHS.
Induction: Assume that 
for an arbitrary y.
We are going to show that 
by the definition of 
.
But .
Hence 
.
But 
by the definition of
.
Hence .
4. 
.