Theorem 2: Let the state vector before step 2a of Grover's algorithm be as follows:

- For the unique marked state
*S*_{m}which satisfies*C*(*S*_{m}) = 1 the amplitude is*k*such that 0 <*k*< 1/ - For each of the remaining (
*N*- 1) states the amplitude is*l*such that*l*> 0

- The change in
*k*,*k*after steps 2a and 2b in Grover's algorithm is bounded below by*k*> - After steps 2a and 2b in Grover's algorithm
*l*> 0

Proof: Let the initial amplitudes be *k* and *l*, let the amplitudes
after the selected phase inversion step 2a be *k*^{'} and *l*^{'}, let
the amplitudes after the inversion about average step 2b be *k*^{''}
and *l*^{''}.

By theorem 1 we know
*k*^{''} = 1 - *k* + 2*l* (note the reversal of terms in the coefficient of
*k*, this is due to the phase inversion of *k* in step 2a), therefore
*k* = *k*^{''} - *k* = - +21 - *l*.
By the assumption
0 < *k* < 1/ and Corollary 1.2 it follows
that
| *l*| > . By assumption *l* is positive, thus
*l* > . Combining this with
*k* = *k*^{''} - *k* = - +21 - *l*. it follows that
*k* > . [Grover96]

To show *l*^{''} positive consider after step 2a of the algorithm,
after the selective phase inversion, but before the inversion about
average. At this point *k*^{'} < 0 and *l*^{'} > 0, since
0 < *k* < and
| *l*| > (from
previous paragraph) that
< .
This means that after step 2a our register is in a state covered by
Corollary 1.1, which states after the inversion about average
operation *l*^{''} will be positive.