RE - Leiden University

RE Axis

S E B x C

TEC xD-

CLAIM : (Ros) OT = Rof Sot )T -Q

41 NOTE "I"

is EQUAliTyOTSL I

NEED "

yep ⇒ XEQyep ⇒ xefaibl Kalb) ERXEQ ⇒ XEP

.

(at A , be B)

RE A xD. * an *

RECALL Rose E A XC , ,

S E B x C Primes" (R' ,s'

)( a ,

b) C- 12*05" ⇐3 To AVOID

T E ( xD CONFUSION ,

- there exists"

a middle"

g ,

THIS HOLDS

i ALWAYSsuch that (a, g) ER & ( 9 ,b )

(l

Fg ED S -T. ( a ,HER AND ( y ,b) ES .

→

WE HAVE TO SET UP OUR SET OF" PREVIOUS KNOWLEDGE "

STATEMENTS

REAXB ,S E Bxc

,TEC xD ; ONE DIRECTION

( a ,'d) E ( Ros) OT ⇒ ( a ,

c- Ro ( Sot )#

( Ca )#

(Ca) ⇒ F y C- E ST

la,9) EROS AND ( Y

,d) ET- -

ku) (Carl

fu) -772 C- Est (a. 2) ER & ( 2,9 ) ES-

(Cme ) ( Cen )

From Kun) , Ken) ,4n) WANT %nYf,( Cl )

RECALL

(Cr) ( a ,c) t Ro ( so -11

1- I e B , faithfully (Dayton : IEEB s# DY

F STEC, ( I , 5) ES & ( 5 ,HTT (Drl (D2)EF5eC st.DKe-

Di

M .IF (Culkin) #m) HOLD THEN 9%4 AND 27-2 SATISFY

Dn'& Df so ( Dn) & (DY ARE TRUE

,AND HENCE 5015144

- ( Ss) ( 541

--

using ( YEC AND (aisle ROTS and ( g.DIET .

ZEB AND ( oh, 2) ER and (2 , 9) ES

"

- -is, ) ( 54

PROVE CD'll :( aid ER & (2.d) c- Sot ( D'll )- -

= Css) ⇒ TRUE REMAINS

(2,41 E so -1 Ig ( 2. g) ES & Ig , d) t 'T

choose g : - y ,SO THEN By (S2) 4154 )

( 2 , g) ES & ( Y,d) ET is TRUE .

So Dh IS TRUE FOR THAT 2,so D1 IS ALSO TRUE

( Ss) ( 541

NEXT WE USE [ YEC AND TEES and THET .]ZEB AND ( A, 2) ER and (2 , 9) ES

"

- -is, ) ( S4

to Prove ④ 4 :( 2 , y ) es & ( y ,dltTBut THESE ARE Exactly (S2 ) & (S4) so Dk is TRUE

,

WITH THAT CHOICE OF Y.

But THEN D2 is TRUE AS SUCH A 4 Exists .

PROVEN : laid ) t ( Ros)oT = ) (a. d) f Ro OT )

REMAINS : ( a ,d ) ERO ( 50T ) ⇒ ( a. d) C- (Ros )oTI# #

(Cal Den-

,

'

#

FYEB,Caister & ( y ,dlESoT

,(Dil f E la ,I ) C- Ros &

/ ( I , d) ET } Daz72 EC ( 9,2 ) C- S 212 , d) ET

'

Ti - l CRI 75 la , 5) ER 415,5 )ts( I cc 's , I

-

-1727172Lc's .ci ,c's ⇒ l

l TAKE 5=51 & 5--2/[Assume ciici.ci ) )

(

y c.'

⇒ Den ,ci⇒Dz2,s'

-

- InDyp remains

LASTLY : Da , ⇐ ' ( a ,2 ) C- Ros- -

⇒ F y la , g) ER & ( 9,2 )tS--

NOTE = NOTE

ca'

a'

so ci.ci ⇒ Den.

so C,

'

. . Cg'

⇒"

ALL"

D 's.

Ro§oT)=(RoS÷ NEXT : in picture.

Pio : REAXB,SEBXC

,TECXD ⇒ (Ros)oT= Robot )

Associativity ONE DIRECTION : la ,d) C- (Ros)oT ⇒ laid ) C- Ro (Sot ),

OF composition

a

faille feedlot ⇒ Is la , b)t.RO#&C2dLET(aihtRos-7F2Ca,2HR,C4HESdog

egg

]⇒ 9 is St

. (2 , g) ES E CY, d) ET ⇒I2 is st ( R & K¥507 ⇒ (a , d) E Ro ( so -1 )