PROPERTIES OF CLASSES OF FUNCTIONS 43 



Aid,,,.,; - e,,,„) S eAixl,,,,, ^::,; (?/'))• 



Since 90^" has the property D there exists a single function n" available 

 as ;Uo' in (rf) and as /i',' in each of the denumerable infinitude of in- 

 stances in (g). Also on account of § 39a4 and 9)J'^'^" there exists a 

 single function m' available as /j.^ in (rf) and as /i„/ in each of the 

 instances in (g) . 



(h) 3m'm" 5 



{h') e : 3 : 3 ml 9 A,',j,,„,. . Kj,,,,,,^. . K',,,,p^,,„y . 3 . 



(/i") ?H . e : 3 : 3 m,,,, 3 K'^,^,,, . K'p^',n ■ K'j,,,,,,;„^^ . 3 . 



A{e^,,,,. - 6,,,.) ^ eAti',,,ix[:„ ip"). 

 (?■)* m; 13:3 m': > vh i m ^ ml' . 3 . 'l.\'„^,,/"^"-". 



(j) ?«r : 3 : 3 m"' > m' 9 -A,',,-,,,/. . Ap^v,/- ■ ■K'^ii'p.'m/" 



.3. .4(0,,.,,. - 0,y„" ) ^ e.4M;>;" (?j")- 

 (A-) p; . p'2 ■ m', . 3 . ( A',',,.,,,/ . a;,,.,,,,.) "' {K',,,'m/ . Ap,.„,,/) 



(.■ft-pj'm,' • -l^pi'm,') \-^pi'm,' • J^pz'm,') • 

 (Ot P'l ■ ]h ■ K'p^'i.i'm," ■ 3 : K'„,'m,, ■ =) • K'j,,-m/' ■ A^,,„,^.< . 



(?n) p[ . p'z . A,',j.,,,.,„../ . 3 . (A^',,.„,_. . K'p„.^^) "' {K'p^'m," • A,',.„,/')- 



(n) A;,-,,,-,,.^,,- .3. A(0,„.,,,- - dp^-pn) ^ eA^^'My' (p")- 



(0) 3 M V" 9 ( f • =) • 3 '". s A;,.,„.,„_ . 3 . a (dp,,p,. - dp,,,., ^ eAjUjvMi/' ip") ) . 



(13)t ^' . A' . ^^i'"''"'--''''' . sp" . gOJ"-": 3 : 0«'-="i''""'.. 3 . ^A-^'Wi-on 



(14) § "^V . A' . sj)}'"'"'-'^'' . S1.V' . aij"°: 3 : 0B"(^"'^'")-. 3 _gAY9i'(«i")_ 



(15) %V . A' . g)?'^'^-'^^' : 3 : ^''*-"''-. 3 .e'"'"'"'. 



" . (9[)^'5[«")i'''^"'''"^^•(9J^'9J^")/'^"'^''='''• 



*By (11) and (8). 

 t By (i). 



t By (12); § 9a5; § 23a4i2; § 39a.5. 

 §By (13); § 9o5; § 23a.3:. 

 II By § 12.1; § 21al, 2, 3, 4; § 39a5, 15; § 29.3. 



