FORMAL REALIZABILITY THEORY — I 267 



Proof: Consider the PR correspondence L defined by Zij)). Then 

 Vi,o = 0, because Z{:p)Q = for every peT/, . Consider the real frame 

 of 13.02. [Z{p)] in this frame takes any of Av+m+i ,•••,/>■» into because 

 these span K^o . Within Ko , [/^(p)] must describe the same operation as 

 the [Zi{p)] of 12.54. Because [Z{y)] is symmetric the lemma follows. 



XVII. THE JUXTAPOSITION OF CORRESPONDENCES 



17.0 Tliis section and the next will consider ways of constructing new 

 correspondences from old. This will provide the basis of the necessity 

 proof of Section 19. 



17.01 It is obvious that if two physical networks are set side by side 

 and their accessible terminals regarded as the terminals of a single 

 larger network, that enlarged network is again a physical network. 

 This is the gist of the present section. 



17.1 Suppose that 



V = Vi e V2 , K = Ki e K2 , 



where K^ = V* and all spaces are real (10.6). Let Ei project on V 

 along V2 (Halmos", par. 33) and E2 = I — Ei project on V2 along Vi . 

 Then E* projects on Ki along Kj , j ^ i (Halmos^, loc. cit.). It is 

 easily verified that Ei = Ei , Ei = Ei , from the analog of 14.02 for 

 dimensionless operators. 



Considering Vi and K, as separate spaces, let Li be a geometrical 

 linear correspondence between them with frequency domain r» , i = 1, 2. 



Consider the correspondence L between V and K defined by 



(i) The frequency domain r^ = rinr2 

 (ii) [v, k]eL(p) if and only if [Eiv, Eik]eLi{p), i = 1, 2. 

 In (ii), of course, we regard EiV and Eik as elements of Vi , Kj . 

 17.11 L so defined is called the juxtaposition of Li and L2 . 



17.2 Lemma: L is PR if and only if each of Li and L2 is PR. 



17.21 Proof of ''if": It is clear that L satisfies PI and P2. Further 

 notation is now simplified if we put Li = M, L-i = N. Consider the 

 manifolds 



v,/ e v^ , v.,/0 e Va-o , k,, e k^ , k.^o e k^o, 



where V m C Vi is the manifold of voltages admitted by Li = M con- 

 sidered as a correspondence between Vi and Ki , and W uo the manifold 



