View of Functional, normal writing language. Proof operators

t70

ììIIIìEIìIìNCES

I{ o f ¡n ân n, \\¡. : Ill[onoloníeeigenscltTflctt cles Steffensen-lterfaln.cns, Acquationes Math., 12, 27-3r (7s75).

'I ^o lrn s on' L' w' nnd S ch o ì 2, I). rì. : on steffensett's DteIr,otr,sIANI J. Nunr. Anaì.,

5,296-302 (1968).

Kantorovich,L.\'.allr.r Aì<iìov,G.p.:Ã-uncrionnlnrrarysis,2ndcd.,Ne¡,yorr<,.

Pergamon Prcss (1982).

stef f enscrì' J. F.: ri¿mar*s ott ileration, Skancr. Aktuar. Tidsriv., r8,64- 72(1g33)..

u I 'nr , ^{S. Y.} : -Ecr¿nsioil:f flllfercil.s ^mcrhod for soluitt¡¡ ttonlineaLí ip'eratot. equalÌons,.

USSR Conrp. ì\Iath., 4, 159-165 (1964).

I'UNCTIONÁ.I_,, NORMAI_, WR,ITING OPERATORS

ZELIIIER AI.EXANDRU (Ctuj_Napoca)

I,ANGUAGE. _PROOF

a,nguage

vill be

done

wiúh the

aid ometrical _essence,

which f calt-,,fiä

ts ngua,ge r{A.fHEÀfATtcA_REvuB _D'ANALYSE

NUMÉRTQUE

BT DE THÉORIB

DE

L'APPROXIMATION

L'ANA,LYSE NUMÉRIQUN

ET LA THÉORIE DE

L,APPROXIMATION Tome

14, No Z, lghó,

ppi.

l7l_t80

t.

^.'IIE FUNCTION¡\L, NORtr{AL MITTNG LAN'UA'D

7.L. Ilte

signs

of

the lønguøge

logical _symbols .4 existential quantifier

B disjunction

^-

C implication

D definition

consúant Þredicates

!,!,

^Q,

^{II, K, p,}

^S,

figures o,1 2,

B) 41

5,6,'

?;

ã,' ^g ä-"å y

L.2.

lormation

_rules

-Tho ^figures and letrers are called

signs.

-

^The numbers are

forméd-àith"" of a silgle

figure, ''gures fonowed

by n -

^{1 signs}

;r",îîu"e

_{ø is}

_the

_number

N. L. VOLLER _12.

t1l t2) t3l t4l t5j t6l

3l

t8l tsl

t 101

[11]

l12l

I 131

I14l

[15 ]

[16

117

Iìcceivcd 24.1I.1985 Mathemalisches Inslttut dcr

Uniaersiläl DÌlsseldorf UníueLsilüIsstrape I D- ⁴⁰⁰⁰ Düsseldorf 1.

U,V L

or of

more

fi-

of

figures. The

772 ZDLMER Af,.

2

liigures succced

in

reverse ord.ei

in

comp_arison

to the

usual rwiting.. rr.or;

:11^nt"'

^{numrrer 12 becomes}

^21y,

^nuniber

^iãz

¡"äåì¡),es

|zryy,

and iso

s an index, an order-number, _a varia_

_are

natural

nonzero numbers.

ber of

(,-I7"

or,

(.D" iff it

is

Ìó[orved

- ^A

^number.is

^an

oÌtlerrnumber

iff it is

follorved

by tt|ú _or

((fu.n,,"

or

^t.'hD,,) _{where /¿}

_is

_{a nurnber,.}

- ^{Variables are}

^sequences.

-

^fndexes

^are

variaibles.

. .

. ^,rr-r,##, ',Ii {:;,Toi"lu r',

^nz _,. ^.

.,

,'h a,re variabres,

then

^((ø, nz . . .

of

a- scquence, .wc tlo noü take

into

udcd

in other

_variables.

is

_a sequence.

P",

_{bcr of}

"?qC", "pqllI", "or^" ^alc

^rvff.

l,he respe<-rtive

opelation,

a,nri

- ^rf

^P

is _is

the ¡rember

of

^(úlrr,.

- ^I1 ni

ø

is

callecl

thc ^",

ú(rßi" are

wff. fn this

case

,, ^{_-.: rf u} ã _: ,:rqii,?t:Piir-,,, ,,n,t¡K,,,

,,nyfr,,,

';#.r"

^a,rc

rvff,

_{cr of} ^"the

;úä;"å

and ₃₁ l,he seconrl j ^firr ùte variables, _Lhert

..rrr2

_{. .}

. _{. ¡}

ür are

called members-, rrl

is

.

e_number .of ^(.D,).

"ptiPq2"

r or

^((f-rr).

its

basis

rn a

wJJ

any index

equal

to a quantifier index is

callecl loose.

index of the

quantifier,

-

^((A)i

^{is a}

^{wff .}

-

Statementas are wff.

-

statements_are numbered. _'r.he order-number

of the

statement.

is

preceded

by a, r, or, (;xióm,ì^ü"o""- or definition).

:3 FUNCTIONAIJ, NORMA¡ _WBrlr¡NG LA¡IGUAGE

773

- ^The

order-numbers

whic,I a.e

included

in a statement must

bo Iess l,tran

the

order_number

of

the

ìiät",r"îîl-

- ^{rn a}

stâtement,

trre

index,es

are,

in-

o'.er of their.

appoar¿ùnce,

the

nrrmtrer:

l, ?, . . ^(ryìtñ

possiblo repeti[ions;.

"r qo^otltlu?*lt"-u-tnt, the grìatest i",I;ï;;;;

_{be equat}

_ro

_{rhe number}

_ _{fn a}

statemenú,

two

clifferen

- ^A 4f ^{is called}

^a

successiv"

^trdexes.

fiel's is the.last _sign of ttre tìâl¿ of

¡re

^quanti-

1,he

forrnula.

sign of

*t""t.1, .fT

^{m be a}

^uff

^auù

ⁿ

^one

of the following

_signs.

ft is

called

thc

nr_if

n is 'últt

^{ot¡ (.1t,}

ot

^(.Zr),

ember of

r

úoqeúher

wiflr

u-,-i1'r iS ,,1ì,,,

, "V"t

^(ú

I'tt. -

with

æ-,

if

e;

is

^,r-Grr, rr7rrr,,..rgrr

6¡ rr¡n.

nbcr, tìre variable_mr-lrui, _of

, åi"ilr""

s sl;atecl

in ^..pq/n.

s sta¿ed

in

^..pqX,,.

stated

in

r', l,hen ,,pqXD is _stated

in

^r,.

re quantifications)

(.p"

rnay be replaced

r

^,,21t,) _then

s

slaterl

in.t.pi.pZ,t.

ts

stâ,üe([

in

^,, pipqZ,, .

s stated

in

^((piPq2tt.

;.

It

inclurled

in

an.y

condition

_þtia,l

in

ù

- ^{rr p is}

^a

^wrr,rhe'

^rrre

"d;,ïlï;,i lï,".íËJ*]ffi:*'.""t

^"'r

yerifv that ¡rcre

exists

oo

^ct

6r,,,1,,,i, 1¡,ì--riä"rfri"..",".

a!

^twitc-

p.

il ;Ht-1"r""'1tÏ1,. t-

n'icf

in p tru'ns inLo '(21'

^¿rnd.

every

^,(7,t,, _lyic

i:::W,:'.o" ^{rtic in}

^{b !}

turns irìto

^rr¡'rr

ald every ((xi, _,,ic in

l¡

! t'rns

cZ

! aild

^(úf!r,'

field. does

not ^orn

^c

!

$,hose

¿

! add

^(31[tt

ol by (tB" or ^{a quautifior}

) FUNCTION.A,L, NORMAL WRMIING LANGUAG{E

1,74 ZELMEn, AT,.

2. PnO()Ì OPtìIl¡tlOItS

4. r75

ú(¡f-lf"

any tinìe it

appears.

wll.

is th ^il: ,1Ë,i:ïHrä#l,iïiLif,::i:ì,Tiî'*1"i,i'f"'il#

sign

of the

formula.

.

_-...p

isasqøstarting*ibh l,hei- quantìfier nicin p(ieN*) if forail

j.

NTr-l; <

i. <

^/c, wheie k

is the n

^oËer

ot-qua"tiri-u'"ì, i" i,-ilr"^¡Jl :Ï:*t1]"1 ,r:t.i" p _{is the}

tas.r^.sign.of rhe

field of

the

j+ ₁ ilr-quanrífie',

lt,rc

rt p,

and

the

/u-th

quantifier nic in p

is

the last

s"ign of p.

1.3.

/

^d,etaíled treatise oJ so,nte erøntples of

writitte

itt, the fultctionul,.

trcrntct L wri,ti,tt(t lan,c¡u,øge,

llhc definil,ion z frorn the

/f-spaces t,heory

nray be

explessccl asr follor,r's.

t-inclucled

in

.I]

if antl

onlr.

if

for-

to 8".

"-4 ^is

point-

stgns repre-

finition,

the

c,

ancl ^ttDtt'

"ø

^belongs

to B"

becomes ^,rBZHrr.

'úrlo.

^ea,ch

noint ø belonging

1.o

Á, t) belongs to B"

llecomes, .L32E3PTTITZ'''.

(r-4

is

point-included

in l/ if

an

lo A,

n

beloigs to

_-Bt' becornes ^r(122

. ^"l-or

^aII

^A,

^B(planes',

pgint

r- belonging

to A, ab GZIG7'),

rvhich

is just

1,he

'r'he a,xiom 2 of 1,_.e /i-spaces

theory

is usually forrnulated as follows,

'(Fol

^cach

straight

linc, ¿r trrere

exists only äne point .r

such thal;,

r is the beginning of

^cli).

We note the variables ¿l and ar J¡¡z rrlrr _and r.2r) . 'l.herefore ((ø

is the

beginning

of

d,r) becoìnes |(ZI(J),.

0"""^fn:,;if;lÌF,.""tv

one poinl' ø such i;hat

r

is the

beginning

_or d))' '(For ^each straight linc ^¿t there exists only

on

e,

beginning. _{91_}

cl')

becomes

tZIUZPI _LSZ,, aïd,

is

of the I12ULP12SZ)).r.ariables according

to

the conditions

imp'o

s.

rn an

operations

on tre

stal,em:nts

have

been noúed

by

ctr, b,

c,.

.,),,_o:.11.!", where!,.,'means

tïaîËåmettring

is addecl

to the

_{ancr '(}

_!"

_rneans

trâtiil";;;Ëo-iäïJ""*,.1t _or

_anorrer

indicat

_{again an¿}

mã¿itiert. r - --*-

_'

2.7.

Operatur nl

.,,rnI

^cxpresses

the function

th¿l,t r.csults fr.oln ^,l¿.

Yelify that

-..

. ^r.-th

rlu¿lntificr,

of

l¿

is

^((1r, ¡¡nil

ln,

t

_S

_{s_ith'thc i_ili} quantificr.

,,1r" ol rn, then

fori

all j, i< j(

^/,,,

'a. rvrite m Lill

to

iJrc l'ielcr

of

f:hc r-i,h

quarrLifitr' _{crusi'.i *}

'l¡.

rv'itc

ûtre intrex r¡nrr

rìrc

crornain

"i ;i;;;-ìh i"äiìi*,]

"r ^,,.

¿'

if

iJre i-th--quan1;ifier of r¿ has

" "onaition, iì -i""iïå

in"

conc'titjo' fol- l<¡tvctl

l)¡' tt-t', if

nol

,

^{g;o bo d.}

d. rvrite the

fiekl

of the ^,i_th

qnantifier

_of

lr.

¿.

\'rite

^ú(dtt.

/.

^{rvrite rvhat is ief1,}

in

tn, alte,- 1,he f_flr cluanfiÏiet,.

!/'

indexes equar

to-the

incre-r

of ilrc

r-1;h

q'antifier

_of

nr

_, oth.errvise

_replace thcrn..

,), l1i,*,

^jr,,,

.-.. jr

nt(l;

_ n:

^e index of tÌre l_ilr cluantifierì of 'li¿. _"

2.2.

Operator

X

tn,nX

infet'frorn

_r¿

_antl

_r¿ l,heir conjunction.

ct.

tvtite

m,.

Ò'

write

r¿ increasing

its

indexcs wil,h flre nurnrrer of qu:r,^tifie.s of ^,¿.

.c.

\ytitetú.,(tt.

îhe t¿sk of this

chapter is language.

Ilor this

reason

the list

ancl

the various

operators

ßa¡r

^s

rator

is an algorithrn r.vith the aicl

2.3. Operator

1,f,

,of

_¿ln

((ftt.

re exisLs rl e

N

suclr Lltt¡b tt¿ is a s¿rn

m a,nd, f h¿r,L bhe fielcl

of this

quanfi- ø.

write the first

member

of ilrat

,,.I1".

ó.

rvrite l'hat is

lefb

in nt

afLi¿t,trÌ,rtr

ii¡".

¿ ! orclcr the indexes.

2.4. _Opera,tor It)

,,_m.iU(í € N*) applics the

s.yrnmetry

to t|c rl-th

operatio¡L ^ú(2,)

or

_"

ÌI" ftom

m.

Verif¡'

that' m, contains a1, least

i

operations ^s(frD

or ,,ll[,,.

r¿.

w'ite

m

tirl to thc first

rncrnber

(e,xcl*si'ely) of the f_tli

opera_

tion

^(tE1)

gy

^tc _{n[tt .}

t76

D.

rvrite thc

sécontl

membeiof thei-th

operation ^...D"

or..MD of

m.

c. wriùe

the first

member

of the i-th

operation

(¡2" or.(M)) of

^m.,

rl. write the i-th operation ..D" or ^,(XIt,

of.lm,.

¿.

write

what is

left in

rn

after ttrc i-th

operation ..n))

ot ^,,M,t

of m..

I ^{! order the}

^indexes.

2.it.

OTterator Jl[

'ntniir . .

.

'i,,lt,.IW

(i

e N8, À; e N,

i,

e N) replaces

in

^n¿

the

stretch of ilre.

i-th-sign by an

ecluivalent

formula'

resurts

frorn

n¿ rvhen r,eplacing

1,he

loose _{tifier of}

tu

by the ir-th

vär,iablõ.

from the

s

do

not countihe

variables rvhich

are loose

i

'om 1,he sl,retch

of

l,he

i-th

sign of m),,

where

I is the

number

of quantifiers of

^m.

Yerify

tha,t

-

^1,he

^r:-th

^si_q*

^{g{*?t i.:} ,..,4,,,,,8,,, ,,C,,,,,I),,, ,,p,,, ,,G,,,,,H,,,,

"I"r "¡¡", ((Xf')i (r'N1',

"P"r ",Stt, tl7rr, ti¡ltt, tt'ytt, ts7tt.

- if ^l, ¡ ^{0, the}

^strel,ch

^{of the i-th}

sign of m, conLains

at

leasl, max

(rr ^ir,.-.

^. r

ià r'ariables rvhich are not looie

inclexcs

of the

quantifier,s.

from the stretch of the r-tli

sign

of

ln.

- ^if

^/r;

^#

^{0, the last /¿} quantifiers _of r¿ are ((Zr)

àttcl r¿

is a

successive, clrranl,ification

starting n'ith

t}-]e

l-lt,),l th quantifier and the field

of-

the l-l;flth quantifier of

r¿ ends

by

,t-ll{".

- il

^lc

: _{0t tt}

_ends

_by

^,(,44tt.

- ^the

^stretch of_th-e'r,-1,h sign of

m

an<\ 1,he

first

member of

thc

ope-

ration

'(JLI_". (indicated_ above)

of

r¿ have

the

sarne number

of

quantifiärs, (we note

this

number

l¡y

p).

ø

!

rvril,e r¿ increasing each

index of

iú

rvith the number of quanti- fiers of

^n¿.

?"1

fot _all

q_e

{!,.2,...,?t\,

^r,cplace

^flre

loosc indexes

of the

q_th

rluantifler, from tlie filst

merrl¡er

of the

operation .,.M,) Trottt ø I

by'the

inclex

of the q-th

cluantifier

from the

stretch

of the i-th

sign on

m.

^"

c! for all je{l,2,.,.,1.:,\,

_r,cplaee

_the

loosc indcxes

of the

]-

- hlitlt

quanbifier from

l¡

! by the

r.r-l'h va,r'ial¡le

frorrr the stretch of t¡e í-th

sign of

?

^{(we do not counl,} thc varialrlcs tha1, are loose indexes of the _euan-

tifiers frorn the strctch of the i-th sign of nt).I)o not .write thc

^{^basis,}

of this quantifier

_{any more,}

r'erify that

ttre resull,

of

1,he su'bstitutioo

,"-

complished

in the

basis (rvithout,

"z)')

represents one or.

t'wo

formulae.

(depentlilg-on the type of

cprantification,-simple

or

conclitioned) n'hicl1 are sl,atecl

in

ln.

Yerify that the first

lnernl¡e,-

or "/4''frclrn c!coincides.n'ith

tìre.

sl;rrltch

of the i-l,h

sign

of

m.

d

^{rvrite r¿ replacing}

thc

sl,retch of the

i-ilr

sign oT nt

by the

seconcÌ member(of ^L(Mt)

from

c !

e

!

order

the

indexes.

2.6,

Operator

Ie

.^nz'i7Q

(ie N*¡

suppresses

the condition of the i-tìr q'antifier or

m.

Verify that

- ^m

conl,ains al, least

i

quantifiers.

- ^{the i-th} quantifier from

r¿

has a

condition.

ø' rwitc

m

tir to

the fielcr of

ilie i-th quantifier

excrusively.

0.

write thc

condition _ancl

the

fielcr

of ilr" f-th

quanl,ifier

of

r¿.

¿.

if the

r,-th cluantifier

of

^/tn

is.ta)) ot

^(,r)), wr.iie

,,x",

_and

if

iü is

t (Z'1

,t rvrite "C",

¿7.

rvrite the

inclex, l,he cromain ancl trre .r-th

quantifiel of

rr,.

¿.

rvrite

rvhal;

is left i' ln

after

the

i_l,h

q..aotifi"".

/! ^order

^{1,he indexes.}

FIINCTIONAI, NONMA¿ WITITING LANGUAGE

2.7. _Operator

N

nrÏ

supposes

ilre contrary of

m,.

Vcrrify

that there

exists

rìo ,,Ct', (.p), tt¡¡n o.

tnemorize m.

Z¡

! rvrite tlie

riegation

of

^nt,.

nlc vI

^1n,.

ZDLMEn' A[,, _Ù ? t77

2.8. Opera,tor c

rm0 recognizes

the

contracliction

of

^ø¿.

Verif¡'

1¡¿tr

-

^{rz¿ ends}

^{by ('xtt}

^{or there exists}

ⁱ

^e

^lN*

^such

^that

^nt, is sclt súarting

with

the^r,--lir Ouqntilier

nic

in.m,, boing

to"ro"J onijiît'ô"u"tifip¡c ^,r/rr

arrd

the field of the i-th

quanbificr _n,ic-tn

r,

"nãr-¡y',ü''ì:--""""''' - ^if

^{rve note}

^{by ¿}

^the

^first

^{membel of}

^the

^{operation ((-ytt inclicatc¿}

a'bove,

there

exists

ito t'C", (tI)), (tflIr'

_,nrc

rtt

1t.

_ ^by

^m_aking

^the

neg'ai,ion

of

n, follurvccl bJ, increasing

the

inclexes

rvith operation'(,Ttt

the number indicatecl above.

of

quantiiiers

of r,

rvc obtain lJie *"ãã"¿

*u*ber

of the ø. rvrite rvhat is

in

t,he tnemory (from a pr.cvious

ìI).

3. ì(-SI"\IJIìS _TIIBOTìY

Ilrom

no\\r on rye shall abbrev

nition _by A., T., D.wlitlen

l,he order-number,

of thc

rcs

¿¡ blanl< space, we

rvrite

the sa_rlr,

on the

follorving

line,

lea follolys

the proof, wriiten from

,^*^-_ï9

TY-t1,,per-anenily

take

care

not to

confound

the sig's of ilro

lTgTrffl ^wlrich

^appeaÌ

^{onty in}

statements,

rvith the notatiõn* oi

iüã

proor

operators, 'ri'hich appear

in the

proofs

of the

theorems.

178 ZDITMEF, AJ-.

3.I.

The

list of

s'l,utements

lL.7 IZH%1\LI{T,BSIBUZlpZ

4.2 IZUIPIZSZ

T.3 7n-nzItzGr2I{ZrBZ

3U2I!2],8T32].352

T

4.4

72n25r211-22rTnr2L7tr

22LrIEXZ1S7,

4.5 I2Iß2VXzSA3PZLPZ

T.6

72U321/X2SI\PZ7PZ

D.7

7272D32H3P:]IIIZM^G7,|GZ

,.8

L282DBZI;BS:I|KZII2GZ\GZ

D.S

72S2DI272DL282DX]/I2G7.TGZ

A.}IY

¹² n 2G129 2 f)z]:g 2 D,Y 7,7e

Z

D

.rry

⁷¹

ry

7 D23 V 2+ U X ⁺ S +1 1t13ñlti

Ii

I ^{2 p} 2rH Z

ttl

^G

Z 4.27Y

I2ÍIN7PZ2G1

'l'.31Y

7LIYOL'KNLSZ

37 y _{N 2Ii1}

I

T 71 S 21 y _It

I

T,y 2

Ii

^{82 I2 S 21 y} tr' 2,f,3 1 /S4 2 lr 2 T C

,f,.41Y

_I2TI1

P T27Y OTÍI Z2G Z 2I'y l' 2 1' 4Iy N I _Q 1 ^-D V 22V I1, 2 tL I ⁽

I

^(-j

T.51y 2ryotr\72D|GZ

+IY 2I'y It \T

Xi

InBt y I22 M 7,t,

T',.67y

1 2

IflSr2Iy

0 p I( zzq Z

:11 y61 _{t: N} I _Q\ \tr 22t/ IT ² + ILX C

I',.7ty 21yOI|I\2D7GZ

67 y _{21Y n} r7,Y811r31 la 722

It

1'f'

?.81L

2lyOr7Ð2D1GZ

51 Y71 y

X\

ⁿ²⁷ y 11 T

X9rn91a

2L2 n[7 T

I',

.97J:

^r 2 I r 13 ¿/,y3/slì 21 y _O p I{ I 2 S

22ty

O p

It IL p].zty

O

r il z

21v F2 T91y _r3r{ 23

Ifi

I +I N 7 Qr ^T X C 6J 3Jr2 K72

K T.02y 2ryo1.r7y7l)

9ly

ztY

I

7 T

^I1

y _{L Ð 7} 3y 71

MI'I'

T

.rzy rzty

o 1182DLGI27 y OI 7 2D

Z

IzY

zLY N T T X 8 92 12 M 7'I' ^N 21 8 ^{7 2}

I{T'I

zIY

r

T T X 7 g 2L2 M ^{L T} 32 T T T 3T S 3 42

s31ZL',ILrQzT2+IOI

7'I 2 7 ^7a Ii- 2'f'

M

^{2 1,} X C 21

y?1

_{1,,y Bt II :lI} y _{12 2,1[ 7}

I, T.22y r2tyoTqzDlq7zry

_OøLDZ

tzy

927 y

n\

^T X22

I(I2

I( 3r Zr

1'atí2't

72g 7 L S 2

Il T.32y

r21 y OntTzDrzty on ^92D MrG 7,

22y 77Q 9 2t y

I

r

r

X22 I{r2 K 3r Z7 T 11 G 7 1' 1 2g

I

^1' lrQ X7 Il,

rzT T.42y

7772DreZ

tl2Y 7 i¡II2 M N778 C 7 I

ngIt2 ttt

_ ^rt

^goes without_saying

that

the lisú

of

sbal,emen{,s

of the

/l-spaces

tìreory

does no1, encl here.

Ilorvevcr, taking into

consicleral,ion th¿r,tt the purpose

of this

pa,pel

is to

plesent onr. Iang"uage,

I

consicler l,he

tist

1,o be

wide

enough.

lj 'i

I FTINCTIONAL, NORMAL WRITÍNG LANGUAG¡' 779

3.2.

Eucnnples

of

proofs

Tlre proof of the

theorern

51Y is +Ly2Iy111L1'XTtEBtyI2ZMIl'

ancl shall be exposed step

by

step. The action

of

each operator

is

shou'n

in

detail.

+7y t2E1Prztyoruzze7,

2TY

^12EN7PZ2GI

27YT

û.

b.

^2G

c.

^2G

d.

^ZGLZHNIPZ

e.

%GTZIINIPZX

r. ^\GLZHNIPZX

a! 2IYOIGI2IYOtr'I{^TII?7'X

h,

! zLvorqtzl yoIHNrPZX 2lyî1I',

e,.

^ZIYOEG

b. 2lyon'G

r:

! 2lYOnG 4lyzLyntL'1'x

0,. IZHIPI2IYOI'EZZGZ

b. IZI:I|PLZLYOFHZ2GZZIY07'e

(i.

12II7P12IYOtr'IIZ2GZ21yOIt'G^'

7

t272D32fI3P31IaZ I[2GZ7GZ 77n

ü,.

b.

^:]2113 P37TIZ

(:.

32H3P3IEZ7272I)

cL.

|t2HBP37EZI272DM

c. 32E3P3IEZI272DM2G7,l

J

I 72EII'\3EZ:|272DM2GZ3GZ

4J Y

zIY II

^{T' X7} IIì:J]-Y I22

M

ct,t.

3+HílP:JúHZ5472DM4GZ5 b

t

I4II1PI-DIIZ|¡L72D ll[\GZl¡GZ

cl 72E7PlztYOEEZZIYOI27LDM

¿t

I 27yOI272D2GZ2lyOnGX

e

!

2[YOtr'772D1GZ)2ITOIG,Y 4r y 2

Iy lll

^T x7

InSty

l'22 M

t I

ü. zlYOmTzDlGZ

b.

2tyon'r72D7q.7,

c

2LTOIL7ZDIG7'

r'hich

is

just

1,hc stalcrrnont o1l l,he theorem 51Y.

it'Ire proof of the theorem 42Y is 42YT\¡IIZMNIIQCTIII9ILZM.

We shall present 1,his

proof

step

b¡'

step indicating 1,he action of each proof operator.

+2Y

IL7ZDIGZ,

7

1272D3283P318'./,M2Gø1G7,

42Y75I72M

LZEIPTZEZ%GZ

+2y75ll2XIN LZHNIPIZHA%AA and

rncrnorize

I2ILIPIZIIZ\GZ

42y75rt2MNLt8 rzHlzENXlIlAzGA

1

È'