Corollary 1.1: We seek to show that after applying *A*, both *k*^{'}
and *l*^{'} are positive, under the following conditions:

Let the state vector for:

- one state have amplitude
*k* - each of the remaining (
*N*- 1) states the amplitude is*l*

*k*and*l*be real*k*be negative, and*l*be positive- <
*N*9

Proof:

First we will show that *k*^{'} is positive:

- From theorem 1 we know
*k*^{'}= - 1*k*+ 2*l*. - By assumption
*k*is negative. Since*N*> 2 by assumption, - 1 is negative. - By assumption
*l*is positive. Since*N*> 2 by assumption, 2 is positive. - Thus the expression for
*k*^{'}is of the form*negative***negative*+*positive***positive*, which must be positive.

Next we will show that *l*^{'} is positive:

- From theorem 1 we know
*l*^{'}=*k*+*l*. - By assumption
<
- For
*N*9, > . Therefore when*N*9: > > , and:*l*^{'}=*k*+*l*>*k*+*l* - Because
*k*is negative and*l*is positive by assumption, = . - Therefore:
*l*^{'}=*k*+*l*>*k*+*l*= ( - 1)*k* - It follows that
*l*^{'}is positive because*k*is positive and -1 > 0 for*N*3 (and by assumption*N*9).