20 S. C.KLEENE

(immediately by#25.1)

(y)n=3fa^,w,

and (with a little work) Cpn,(y), and

hence by#31•3 Cp(y); forillustration, we dothework to establish thesixth

of theeight conjunctands in Cp0b(y), namely (y)Q=«0,(y)Q

Q 1

,A,B,(y)Q .

First, (y)QQ = f [by(y)Q=f,a,b,w with *25.l] = (y)QQ [by (y)Q=f,a,b,w]

= 0 (y)0,o'l tb^ ty)0=«0,(y)0iOA,... inCpQb(y)J/Hence ( ^ ^ ( y ^ .

So (a) f=0,(y)Q

Q

X. Second, (y)Q± - a [by (y)0=f,a,b,w]= (y)Qx [by

(y)Q=f,a,b,w]= A [by (y)0=«0,(y)0^0^1,A,...] = 1 1 ^ ) p^l,! [by

definition!. So(y)0jlj0=(y\1)0 ^

d V i

i(y)

0 ) 1

// ^ ' ^ l

°'1±'

whence (b) a =TJ.M p^O^A =!!,,s V^0'1'1 fa *B±9 l =

ni(y)n , A 7 °'1,i=X' Th±rd' (^o,2=^ so (bV(^o,2,o ^

w

(EL (y)0oi . , _

T-T

(b)tr ,

V i

i(b)

0

Pi_ ^ i " • Hence ( c ) b = | l ^ ^ i te*™d =

Fl",l.

n m9

l - T I w - x = B. Fourth, (d) w=(?)0 .

i(b)px!y)°2,i_[*

0

_ ^^0,2,0p.(7;°»2»i

Using (a)-(d) in(y)Q=f,a,b,w, we get (y)0=«0,(y)0

Q 1

,A,B,(y)Q , as

required. — How weuse&- and 3-introd.; andcomplete the 3u- and 3v-elims.

CASE 7: Cp?(y). Now0 (y)Q1

Q

= (a)Qand0 (y)Q2

Q

= (b)Qandhence

so by (b^lKb^ and*29.1, ((b)1)(a) =((£)-, )(a) * Let y=fa'b,w. Then

(y)0=f,a,b,wj and ( ( y ) ^ ^ ) ^

x x

= ^iV^ = ((b)l)(a)1'so

((7)0,2,l)(y)0jljl^«*

^ W ^ ^ /

1 =

^

WWo,!,!*1=

^

[by Cp„(y)] = w [by(y)Q=f,a,b,w]. Now we easily obtain Cp„(y), whence Cp(y).

CASE 8: Cpg(y). Then (a/^ (y).jy and (tOCp((y)1) and[using (y)QQ -^(f-)-^

(y)0jl=a, (y)o,o,2=(f52, Vj(y)0j2,r(b)o ^

*ml%

and (y)o,3=w]

(c) (y),n=(f),,a,C,w where

^" • ^ o ,n

( b )

i^

(b)i

rr

(b)i

Let C beCwith b substituted forb# Using *3.9 with --18.5 and*B6, each of

the four factors in C or(J,andhence by*129 etc. each of C andG, is 0;

so using first -19.11 andthen *19.9 (with *18.1), -19.10, --19.13, -19.14: