FORMAL REALIZABILITY THEORY — I 249 



by the formulas of 10.5. This means that [L] and [L]i are related like the 

 [L] and [L „ ] of 9.2. The 2/i-pole associated with [L]i therefore is a Cauer 

 equivalent of that associated with [L]. 



11.23 The observation of 11.22, combined with (ii) of 11.13, gives an 

 alternative proof of 9.31. This proof is deceptively free of calculation, 

 but of course the calculations are concealed in the extensive geometrical 

 developments of Section 10, many of which are there offered on faith. 



XII. THE FUNDAMENTAL LEMMA 



12.0 This section is devoted to the statement, and the proof in part, of a 

 lemma which, on the face of it, looks like an exercise in manipulating the 

 postulates. In fact, the content of the lemma, and most of the details of 

 its proof, are essential in what follows. To postpone them would force us 

 into needless duplication of effort. 



Lemma: Let L be a geometrical linear correspondence satisfying "1, 

 P2, P4, Po(I), P6(I), P7(I) and the following w^eak form of P3(I): 



P3'(I): If per, , then K,(p) = K, 3 (V,.o(p))". 



Then there is a frequency domain t'l C Tl , differing from Pz, by a 

 finite set, such that L satisfies all of the postulates for peVL- 



The statement of the dual result is evident and will be omitted. 



The proof that L satisfies P3 will be given in this section. Verification 

 of the remaining postulates will follow in paragraph 16.6. 



We assume that the properties of positive real (PR) functions are 

 known. They are summarized for later use in Section 15. We make 

 occasional advance references thereto. 



To the proof: 



12.01 First, Kl is a real manifold and for peTL 



K. C (V.o(p))". (1) 



This, with P3'(I), gives P3(I) for L. 



Proof: Kl is real, as in 7.51. Consider no>v^ a pePz, c, ^tl a vcVlo(p) ; 

 then [v, 0]eL{p). Consider any real jtK^ ; then there is a utV^ip) 

 such that [u, j]eL(p). Now and j are real. Hence by P5(I) 



(v, j) = {u, 0) = 0. 



Therefore any real jeKt has a vanishing scalar product with every 

 reYLoip). Since Kl is real, it is spanned by real j and (1) follows. 



12.1 By the dual of 7.7, if we know that 



[v, k]eL{p), 



