FORMAL REALIZABILITY THEORY — II 587 



every successive 6 > 1 such that ft > 0, 



7i + • • • + 7!> > 7i + • • • + 76 • 



This inequahty is now not altered if non-negative numbers are added 

 to its loft member and zeros to its right member. Hence it holds for all 

 b, 1 < b < n, and 



S{Z, po) > S(FZ, po). (6) 



This is the first claim of the theorem. 



Now if F(p) is non-singular at po , then F (p) is rational, and finite 

 at Po . Hence by what is already proved, 



S{FZ, Po) > S{F~\FZ), Po). 



This last array is just S{Z, po). Hence we have (6) and its reverse, and 

 the theorem is proved. 



5.27 Theorem: \{ po 9^ °o and 



z{p) = ZM + z,(p), 



where Z-^ip) is finite at po , then 



S(Z, Po) = S{Z, , Po). 

 The proof of this depends upon the foUoAving lemma. 



5.28 Lemma: Let Z*{p) be such that at p = po ?^ °° its only elements 

 having poles are on the main diagonal. Let — fi , — £2 , • • • be the ex- 

 ponents of (p — Po) in the diagonal elements of Z*(p), so enumerated 

 that 



-\- £1 > -\- £2 > •■ ' > + £n ■ 



Let —ti, —&2, • ■ • , — f „ be the exponents of (p — po) in the successive 

 diagonal elements of the normal form of Z*{p). Then if tb > we have 



f 1 + • • • + £b > £1+ ' • • + f 6 • 



Proof: There exist constant non-singular matrices F, G such that 

 FZ*G has the same rows and columns as Z* so permuted that the diag- 

 onal elements of FZ*G are arranged in the order of ascending powers of 

 (p — Po), the highest order pole l)eing in the first position. Since the 

 normal forms of Z* and FZ*G are identical, it suffices to consider Z* it- 

 self to be in this form. 



Let \p = \pz'{p). Now xpZ* has its diagonal elements in the order of 

 increasing posit i\'e power of (p — po). Furthermore, any off-diagonal 

 element of ypZ* has (p — po)'^ as a factor. 



