FORMAL REALIZABILITV THEORY — II 589 



By induction on b, then, 



Sk = £k 

 for k = 1, 2, etc. until such k tliat both are negative. Therefore 



7a = sup (ft ,0) = Tfc = sup (f'k , 0) 



for all /,'=], 2, •••, n. That is, 



S(Z, po) = S(Zi , po), 



Q.E.D. 



").29 Theorem: Let Z(p) be such that a.t p = pn 9^ <^ its only elements 

 having poles lie on the main diagonal. Let ai , a-> , • • • , cr„ be the orders 

 of these poles, so enumerated that 



CTl > O") > • • • > (T„ . 



Then 



S(Z, Po) = [o-i , 0-0 , • • • , (T,J. 



Proof: We write 



Z(?>) = Z*{p) + Z,(p), 



where Z*(p) is diagonal, having exactly the diagonal elements of Z(p). 

 By 5.27, 



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



Now Z*(p) falls under 5.28, but is diagonal in addition. Li the proof 

 of 5.28, therefore, it is exactly the principal minors of if/Z* which have 

 the lowest exponents for (p — po), since all non-principal minors vanish 

 and have zeros of arbitrary order at p = po • Furthermore, (7) is exactly 

 the least exponent of (p — po) in any 6-rowed minor of xpZ* since the 

 principal minors are simple products. Hence (7) and (8) are equal, for 

 any 6 = 1,2, • • • , n. Therefore the exponents in the normal form 

 of Z* are exactly those of Z* and 



S{Z, Po) = S{Z*, Po) = kl , (72 , • • • , (tJ. 



Q.E.D. 



5.3. Definition: Let 



p = 3-(«) = ^ 

 yq -\- 8 



be a non-singular bi-rational transformation from the 5-sphere to the 



