FORMAL REALIZABILITY THEORY — ^I 257 



the correspondeiiee of 12.32 and 12.54 between V2 and K> is bi-uniqiie 

 for any peF/, . This extends 12.31 to any peF;, . 



12.57 If m - 0, i.e., if V, = K, = 0, then V,,, = (K,,,,)' and the fact 

 that L satisfies all the i)()stulates is trix'ial l)eeause all scalar prtxhicts 

 (r, A') for reV;, = Vm and A'eK/, = Klo are zcn-o. If )n ^ 0, we have yet 

 to show that L satisfies P5(A), P(KA), P7(A). 



12.6 8in('(> now L satisfies P3, 7.7 as gi\'en is ai^plicable and we find 

 (with 12.1) that if peV',. and 



then {v, k) is fixed by either v or /,-. Furthermore, 



(V, k) = (V + To , k + /,•„) 



for any roeV/ji , koeKio . 



12.61 If perl and [v, k]eL(p), then reV;, , A'eK;. . By (30), (31), and 

 (16), therefore, there exist roeVz.n , koeKu< such that u = v — VoeV^ , 

 j = (k - ko)eK.2 . Then by P2 



[>',j]eUp). (32) 



By 12.6, theji, any value assumed by a scalar product {v, k) with 

 [r, l:]eL{p) is also assumed by a product (n, j), where (32) holds and 

 ;/eV, , jeK. . 



XIII. SUFFICIEXCY OF THE POSTULATES 



13.0 We suppose that L satisfies the postulates of 12.0. Then the results 

 of Section 12 are applicable. The ones of first importance are contained 

 in the facts from (15), (30) and (31), that 



V;. = Vz.0 e V, , 



K, = Ko e K,o , 



where the choice of Ko was governed only by the requirement that the 

 second of these formulae hold. 



13.01 Considering K2 and V2 as separate spaces, V2 = K2 by 10.6. 

 Let .1/ be the geometrical linear correspondence between them with 

 frequency domain T,, and pairs described by 12.31 and 12.56 (or 12.54). 

 That is, as vectors in V2 and K2 



[V, k]eM^) 



