FORMAL REALIZABILITY THEORY — I 251 



12.13 We wish now to show that Ki C K/,o(/)). Consider a real jeK^ and 

 a real keKi . Let 



(4) 



Hp),jhL(p), 



Hp), k]eL(p). 

 Then, given any real X, by P2 



[u{p) -^Xiip),j -{-U]eL{p). 



Then, because keKi , 



{u + Xv,j + XA-) = ((/,./) + \{v,j) + X(w, k). 



Since j and k are real, by P5(I) this can be written 



(w + Xr, ./ + X/o) = (uj) +2\{v,j). (5) 



Choose any pi such that Re(pi) > 0. Then P7(I) implies that the left 

 side of (5) has a non-negative real part at p = pi . The right side, by 

 suitable choice of X, can have any chosen real part unless 



Re(Kpi),i) = 0. (6) 



Hence P7(I) implies (6). Now (v{p), j) is a rational function, by P6(I) 

 applied to the other members of (5). By (6), this rational function has a 

 vanishing real part throughout the right half p-plane. Hence it is an 

 imaginary constant: 



(v(p)J) ^ iO" (7) 



Then 



iv(p),j) = (.v(p),j) = -ia. (8) 



But [v(p), k]eL(p), so [v(p), k]eL(p) by P4. Since also [v(p), k]eL(p), 

 by 12.1, we have from (8) that 



{v(p),j) = -ia. 



Comparing this with (7) written for p, we have a = and 



iv(p)J) = for peT,. (9) 



Now v(p) was determined by (4) wherein k is real. For any /ceKi , 

 k = ki + ik2 , where ki and /jo are real and in Ki (12.11). A correspond- 

 ing v(p) satisfying (4) can be written 



v(p) = vi{p) + iViip), (10) 



