:: Non negative real numbers. Part I
:: by Andrzej Trybulec
::
:: Received March 7, 1998
:: Copyright (c) 1998 Association of Mizar Users

begin

definition

func DEDEKIND_CUTS -> Subset-Family of RAT+ equals :: ARYTM_2:def 1
{ A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) ) } \ {RAT+};
coherence
{ A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) ) } \ {RAT+} is Subset-Family of RAT+

proof end;

end;

:: deftheorem defines DEDEKIND_CUTS ARYTM_2:def 1 :
DEDEKIND_CUTS = { A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) ) } \ {RAT+};

registration

cluster DEDEKIND_CUTS -> non empty ;
coherence
not DEDEKIND_CUTS is empty

proof end;

end;

definition

func REAL+ -> set equals :: ARYTM_2:def 2
(RAT+ \/ DEDEKIND_CUTS) \ { { s where s is Element of RAT+ : s < t } where t is Element of RAT+ : t <> {} } ;
coherence
(RAT+ \/ DEDEKIND_CUTS) \ { { s where s is Element of RAT+ : s < t } where t is Element of RAT+ : t <> {} } is set ;

end;

:: deftheorem defines REAL+ ARYTM_2:def 2 :
REAL+ = (RAT+ \/ DEDEKIND_CUTS) \ { { s where s is Element of RAT+ : s < t } where t is Element of RAT+ : t <> {} } ;

set IR = { A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) ) } ;

set RA = { { s where s is Element of RAT+ : s < t } where t is Element of RAT+ : t <> {} } ;

Lm1: for x, y being set holds not [x,y] in { A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) ) }

proof end;

Lm2: RAT+ misses { { s where s is Element of RAT+ : s < t } where t is Element of RAT+ : t <> {} }

proof end;

theorem Th1: :: ARYTM_2:1

RAT+ c= REAL+ by Lm2, XBOOLE_1:7, XBOOLE_1:86;

RAT+ \/ DEDEKIND_CUTS c= RAT+ \/ { A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) ) }
by XBOOLE_1:9;

then Lm3: REAL+ c= RAT+ \/ { A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) ) }
by XBOOLE_1:1;

REAL+ = RAT+ \/ (( { A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) ) } \ {RAT+}) \ { { s where s is Element of RAT+ : s < t } where t is Element of RAT+ : t <> {} } )
by Lm2, XBOOLE_1:87;

then Lm4: DEDEKIND_CUTS \ { { s where s is Element of RAT+ : s < t } where t is Element of RAT+ : t <> {} } c= REAL+
by XBOOLE_1:7;

Lm5: omega c= ( { [c,d] where c, d is Element of omega : ( c,d are_relative_prime & d <> {} ) } \ { [k,1] where k is Element of omega : verum } ) \/ omega
by XBOOLE_1:7;

theorem :: ARYTM_2:2

omega c= REAL+ by Lm5, Th1, XBOOLE_1:1;

registration

cluster REAL+ -> non empty ;
coherence
not REAL+ is empty by Th1;

end;

definition

let x be Element of REAL+ ;
func DEDEKIND_CUT x -> Element of DEDEKIND_CUTS means :Def3: :: ARYTM_2:def 3
ex r being Element of RAT+ st
( x = r & it = { s where s is Element of RAT+ : s < r } ) if x in RAT+
otherwise it = x;
existence
( ( x in RAT+ implies ex b₁ being Element of DEDEKIND_CUTS ex r being Element of RAT+ st
( x = r & b₁ = { s where s is Element of RAT+ : s < r } ) ) & ( not x in RAT+ implies ex b₁ being Element of DEDEKIND_CUTS st b₁ = x ) )

proof end;

uniqueness
for b₁, b₂ being Element of DEDEKIND_CUTS holds
( ( x in RAT+ & ex r being Element of RAT+ st
( x = r & b₁ = { s where s is Element of RAT+ : s < r } ) & ex r being Element of RAT+ st
( x = r & b₂ = { s where s is Element of RAT+ : s < r } ) implies b₁ = b₂ ) & ( not x in RAT+ & b₁ = x & b₂ = x implies b₁ = b₂ ) ) ;
consistency
for b₁ being Element of DEDEKIND_CUTS holds verum ;

end;

:: deftheorem Def3 defines DEDEKIND_CUT ARYTM_2:def 3 :
for x being Element of REAL+
for b₂ being Element of DEDEKIND_CUTS holds
( ( x in RAT+ implies ( b₂ = DEDEKIND_CUT x iff ex r being Element of RAT+ st
( x = r & b₂ = { s where s is Element of RAT+ : s < r } ) ) ) & ( not x in RAT+ implies ( b₂ = DEDEKIND_CUT x iff b₂ = x ) ) );

theorem :: ARYTM_2:3

for y being set holds not [{},y] in REAL+

proof end;

Lm6: for s, t being Element of RAT+ st ( for r being Element of RAT+ holds
( r < s iff r < t ) ) holds
s = t

proof end;

definition

let x be Element of DEDEKIND_CUTS ;
func GLUED x -> Element of REAL+ means :Def4: :: ARYTM_2:def 4
ex r being Element of RAT+ st
( it = r & ( for s being Element of RAT+ holds
( s in x iff s < r ) ) ) if ex r being Element of RAT+ st
for s being Element of RAT+ holds
( s in x iff s < r )
otherwise it = x;
uniqueness
for b₁, b₂ being Element of REAL+ holds
( ( ex r being Element of RAT+ st
for s being Element of RAT+ holds
( s in x iff s < r ) & ex r being Element of RAT+ st
( b₁ = r & ( for s being Element of RAT+ holds
( s in x iff s < r ) ) ) & ex r being Element of RAT+ st
( b₂ = r & ( for s being Element of RAT+ holds
( s in x iff s < r ) ) ) implies b₁ = b₂ ) & ( ( for r being Element of RAT+ holds
not for s being Element of RAT+ holds
( s in x iff s < r ) ) & b₁ = x & b₂ = x implies b₁ = b₂ ) )

proof end;

existence
( ( ex r being Element of RAT+ st
for s being Element of RAT+ holds
( s in x iff s < r ) implies ex b₁ being Element of REAL+ ex r being Element of RAT+ st
( b₁ = r & ( for s being Element of RAT+ holds
( s in x iff s < r ) ) ) ) & ( ( for r being Element of RAT+ holds
not for s being Element of RAT+ holds
( s in x iff s < r ) ) implies ex b₁ being Element of REAL+ st b₁ = x ) )

proof end;

consistency
for b₁ being Element of REAL+ holds verum ;

end;

:: deftheorem Def4 defines GLUED ARYTM_2:def 4 :
for x being Element of DEDEKIND_CUTS
for b₂ being Element of REAL+ holds
( ( ex r being Element of RAT+ st
for s being Element of RAT+ holds
( s in x iff s < r ) implies ( b₂ = GLUED x iff ex r being Element of RAT+ st
( b₂ = r & ( for s being Element of RAT+ holds
( s in x iff s < r ) ) ) ) ) & ( ( for r being Element of RAT+ holds
not for s being Element of RAT+ holds
( s in x iff s < r ) ) implies ( b₂ = GLUED x iff b₂ = x ) ) );

Lm7: for r being Element of RAT+
for B being set st B in DEDEKIND_CUTS & r in B holds
ex s being Element of RAT+ st
( s in B & r < s )

proof end;

Lm8: {} in DEDEKIND_CUTS

proof end;

Lm9: DEDEKIND_CUTS /\ RAT+ = {{}}

proof end;

Lm10: for x being Element of REAL+ holds
( DEDEKIND_CUT x = {} iff x = {} )

proof end;

Lm11: for A being Element of DEDEKIND_CUTS holds
( GLUED A = {} iff A = {} )

proof end;

Lm12: for A being Element of DEDEKIND_CUTS holds DEDEKIND_CUT (GLUED A) = A

proof end;

Lm13: for x, y being set st x in { A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) ) } & y in { A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) ) } & not x c= y holds
y c= x

proof end;

definition

let x, y be Element of REAL+ ;
pred x <=' y means :Def5: :: ARYTM_2:def 5
ex x9, y9 being Element of RAT+ st
( x = x9 & y = y9 & x9 <=' y9 ) if ( x in RAT+ & y in RAT+ )
x in y if ( x in RAT+ & not y in RAT+ )
not y in x if ( not x in RAT+ & y in RAT+ )
otherwise x c= y;
consistency
( ( x in RAT+ & y in RAT+ & x in RAT+ & not y in RAT+ implies ( ex x9, y9 being Element of RAT+ st
( x = x9 & y = y9 & x9 <=' y9 ) iff x in y ) ) & ( x in RAT+ & y in RAT+ & not x in RAT+ & y in RAT+ implies ( ex x9, y9 being Element of RAT+ st
( x = x9 & y = y9 & x9 <=' y9 ) iff not y in x ) ) & ( x in RAT+ & not y in RAT+ & not x in RAT+ & y in RAT+ implies ( x in y iff not y in x ) ) ) ;
connectedness
for x, y being Element of REAL+ st ( ( x in RAT+ & y in RAT+ & ( for x9, y9 being Element of RAT+ holds
( not x = x9 or not y = y9 or not x9 <=' y9 ) ) ) or ( x in RAT+ & not y in RAT+ & not x in y ) or ( not x in RAT+ & y in RAT+ & y in x ) or ( ( not x in RAT+ or not y in RAT+ ) & ( not x in RAT+ or y in RAT+ ) & ( x in RAT+ or not y in RAT+ ) & not x c= y ) ) holds
( ( y in RAT+ & x in RAT+ implies ex x9, y9 being Element of RAT+ st
( y = x9 & x = y9 & x9 <=' y9 ) ) & ( y in RAT+ & not x in RAT+ implies y in x ) & ( not y in RAT+ & x in RAT+ implies not x in y ) & ( ( not y in RAT+ or not x in RAT+ ) & ( not y in RAT+ or x in RAT+ ) & ( y in RAT+ or not x in RAT+ ) implies y c= x ) )

proof end;

end;

:: deftheorem Def5 defines <=' ARYTM_2:def 5 :
for x, y being Element of REAL+ holds
( ( x in RAT+ & y in RAT+ implies ( x <=' y iff ex x9, y9 being Element of RAT+ st
( x = x9 & y = y9 & x9 <=' y9 ) ) ) & ( x in RAT+ & not y in RAT+ implies ( x <=' y iff x in y ) ) & ( not x in RAT+ & y in RAT+ implies ( x <=' y iff not y in x ) ) & ( ( not x in RAT+ or not y in RAT+ ) & ( not x in RAT+ or y in RAT+ ) & ( x in RAT+ or not y in RAT+ ) implies ( x <=' y iff x c= y ) ) );

notation

let x, y be Element of REAL+ ;
antonym y < x for x <=' y;

end;

Lm14: for x9, y9 being Element of RAT+
for x, y being Element of REAL+ st x = x9 & y = y9 holds
( x <=' y iff x9 <=' y9 )

proof end;

Lm15: for B being set st B in DEDEKIND_CUTS & B <> {} holds
ex r being Element of RAT+ st
( r in B & r <> {} )

proof end;

Lm16: for r, s being Element of RAT+
for B being set st B in DEDEKIND_CUTS & r in B & s <=' r holds
s in B

proof end;

Lm17: for B being set st B in DEDEKIND_CUTS & B <> {} holds
{} in B

proof end;

Lm18: for r being Element of RAT+
for B being set st B in DEDEKIND_CUTS \ { { s where s is Element of RAT+ : s < t } where t is Element of RAT+ : t <> {} } & not r in B & B <> {} holds
ex s being Element of RAT+ st
( not s in B & s < r )

proof end;

Lm19: for x being Element of REAL+ st DEDEKIND_CUT x in { { s where s is Element of RAT+ : s < t } where t is Element of RAT+ : t <> {} } holds
x in RAT+

proof end;

Lm20: for x, y being Element of REAL+ st DEDEKIND_CUT x c= DEDEKIND_CUT y holds
x <=' y

proof end;

Lm21: for x, y being Element of REAL+ st x <=' y & y <=' x holds
x = y

proof end;

Lm22: for x, y being Element of REAL+ st DEDEKIND_CUT x = DEDEKIND_CUT y holds
x = y

proof end;

Lm23: for x being Element of REAL+ holds GLUED (DEDEKIND_CUT x) = x

proof end;

definition

let A, B be Element of DEDEKIND_CUTS ;
func A + B -> Element of DEDEKIND_CUTS equals :Def6: :: ARYTM_2:def 6
{ (r + s) where r, s is Element of RAT+ : ( r in A & s in B ) } ;
coherence
{ (r + s) where r, s is Element of RAT+ : ( r in A & s in B ) } is Element of DEDEKIND_CUTS

proof end;

commutativity
for b₁, A, B being Element of DEDEKIND_CUTS st b₁ = { (r + s) where r, s is Element of RAT+ : ( r in A & s in B ) } holds
b₁ = { (r + s) where r, s is Element of RAT+ : ( r in B & s in A ) }

proof end;

end;

:: deftheorem Def6 defines + ARYTM_2:def 6 :
for A, B being Element of DEDEKIND_CUTS holds A + B = { (r + s) where r, s is Element of RAT+ : ( r in A & s in B ) } ;

Lm24: for B being set st B in DEDEKIND_CUTS holds
ex r being Element of RAT+ st not r in B

proof end;

definition

let A, B be Element of DEDEKIND_CUTS ;
func A *' B -> Element of DEDEKIND_CUTS equals :: ARYTM_2:def 7
{ (r *' s) where r, s is Element of RAT+ : ( r in A & s in B ) } ;
coherence
{ (r *' s) where r, s is Element of RAT+ : ( r in A & s in B ) } is Element of DEDEKIND_CUTS

proof end;

commutativity
for b₁, A, B being Element of DEDEKIND_CUTS st b₁ = { (r *' s) where r, s is Element of RAT+ : ( r in A & s in B ) } holds
b₁ = { (r *' s) where r, s is Element of RAT+ : ( r in B & s in A ) }

proof end;

end;

:: deftheorem defines *' ARYTM_2:def 7 :
for A, B being Element of DEDEKIND_CUTS holds A *' B = { (r *' s) where r, s is Element of RAT+ : ( r in A & s in B ) } ;

definition

let x, y be Element of REAL+ ;
func x + y -> Element of REAL+ equals :Def8: :: ARYTM_2:def 8
x if y = {}
y if x = {}
otherwise GLUED ((DEDEKIND_CUT x) + (DEDEKIND_CUT y));
coherence
( ( y = {} implies x is Element of REAL+ ) & ( x = {} implies y is Element of REAL+ ) & ( not y = {} & not x = {} implies GLUED ((DEDEKIND_CUT x) + (DEDEKIND_CUT y)) is Element of REAL+ ) ) ;
consistency
for b₁ being Element of REAL+ st y = {} & x = {} holds
( b₁ = x iff b₁ = y ) ;
commutativity
for b₁, x, y being Element of REAL+ st ( y = {} implies b₁ = x ) & ( x = {} implies b₁ = y ) & ( not y = {} & not x = {} implies b₁ = GLUED ((DEDEKIND_CUT x) + (DEDEKIND_CUT y)) ) holds
( ( x = {} implies b₁ = y ) & ( y = {} implies b₁ = x ) & ( not x = {} & not y = {} implies b₁ = GLUED ((DEDEKIND_CUT y) + (DEDEKIND_CUT x)) ) ) ;
func x *' y -> Element of REAL+ equals :: ARYTM_2:def 9
GLUED ((DEDEKIND_CUT x) *' (DEDEKIND_CUT y));
coherence
GLUED ((DEDEKIND_CUT x) *' (DEDEKIND_CUT y)) is Element of REAL+ ;
commutativity
for b₁, x, y being Element of REAL+ st b₁ = GLUED ((DEDEKIND_CUT x) *' (DEDEKIND_CUT y)) holds
b₁ = GLUED ((DEDEKIND_CUT y) *' (DEDEKIND_CUT x)) ;

end;

:: deftheorem Def8 defines + ARYTM_2:def 8 :
for x, y being Element of REAL+ holds
( ( y = {} implies x + y = x ) & ( x = {} implies x + y = y ) & ( not y = {} & not x = {} implies x + y = GLUED ((DEDEKIND_CUT x) + (DEDEKIND_CUT y)) ) );

:: deftheorem defines *' ARYTM_2:def 9 :
for x, y being Element of REAL+ holds x *' y = GLUED ((DEDEKIND_CUT x) *' (DEDEKIND_CUT y));

theorem Th4: :: ARYTM_2:4

for x, y being Element of REAL+ st x = {} holds
x *' y = {}

proof end;

theorem :: ARYTM_2:5

canceled;

theorem Th6: :: ARYTM_2:6

for x, y being Element of REAL+ st x + y = {} holds
x = {}

proof end;

Lm25: for A, B, C being Element of DEDEKIND_CUTS holds A + (B + C) c= (A + B) + C

proof end;

Lm26: for A, B, C being Element of DEDEKIND_CUTS holds A + (B + C) = (A + B) + C

proof end;

Lm27: for A, B being Element of DEDEKIND_CUTS holds
( not A + B = {} or A = {} or B = {} )

proof end;

theorem Th7: :: ARYTM_2:7

for x, y, z being Element of REAL+ holds x + (y + z) = (x + y) + z

proof end;

theorem :: ARYTM_2:8

{ A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) ) } is c=-linear

proof end;

Lm28: for e being set st e in REAL+ holds
e <> RAT+

proof end;

Lm29: for r, s being Element of RAT+
for B being set st B in { A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) ) } & r in B & s <=' r holds
s in B

proof end;

Lm30: for y, x being Element of REAL+ st y < x holds
ex z being Element of REAL+ st
( z in RAT+ & z < x & y < z )

proof end;

Lm31: for x, y, z being Element of REAL+ st x <=' y & y <=' z holds
x <=' z

proof end;

theorem :: ARYTM_2:9

for X, Y being Subset of REAL+ st ex x being Element of REAL+ st x in Y & ( for x, y being Element of REAL+ st x in X & y in Y holds
x <=' y ) holds
ex z being Element of REAL+ st
for x, y being Element of REAL+ st x in X & y in Y holds
( x <=' z & z <=' y )

proof end;

Lm32: one = 1
;

Lm33: {} = {}
;

Lm34: for A, B being Element of DEDEKIND_CUTS st A + B = A & A <> {} holds
B = {}

proof end;

Lm35: for x, y being Element of REAL+ st x + y = x holds
y = {}

proof end;

Lm36: for A, B being Element of DEDEKIND_CUTS st A <> {} & A c= B & A <> B holds
ex C being Element of DEDEKIND_CUTS st A + C = B

proof end;

Lm37: for x, y being Element of REAL+ st x <=' y holds
DEDEKIND_CUT x c= DEDEKIND_CUT y

proof end;

theorem Th10: :: ARYTM_2:10

for x, y being Element of REAL+ st x <=' y holds
ex z being Element of REAL+ st x + z = y

proof end;

theorem Th11: :: ARYTM_2:11

for x, y being Element of REAL+ ex z being Element of REAL+ st
( x + z = y or y + z = x )

proof end;

theorem Th12: :: ARYTM_2:12

for x, y, z being Element of REAL+ st x + y = x + z holds
y = z

proof end;

Lm38: for A, B, C being Element of DEDEKIND_CUTS holds A *' (B *' C) c= (A *' B) *' C

proof end;

Lm39: for A, B, C being Element of DEDEKIND_CUTS holds A *' (B *' C) = (A *' B) *' C

proof end;

theorem :: ARYTM_2:13

for x, y, z being Element of REAL+ holds x *' (y *' z) = (x *' y) *' z

proof end;

Lm40: for x, y being Element of REAL+ holds
( not x *' y = {} or x = {} or y = {} )

proof end;

Lm41: for A, B, C being Element of DEDEKIND_CUTS holds A *' (B + C) = (A *' B) + (A *' C)

proof end;

theorem :: ARYTM_2:14

for x, y, z being Element of REAL+ holds x *' (y + z) = (x *' y) + (x *' z)

proof end;

one in RAT+
;

then reconsider rone = one as Element of REAL+ by Th1;

Lm42: for B being set st B in { A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) ) } & B <> {} holds
ex r being Element of RAT+ st
( r in B & r <> {} )

proof end;

Lm43: for A being Element of DEDEKIND_CUTS st A <> {} holds
ex B being Element of DEDEKIND_CUTS st A *' B = DEDEKIND_CUT rone

proof end;

theorem :: ARYTM_2:15

for x being Element of REAL+ st x <> {} holds
ex y being Element of REAL+ st x *' y = one

proof end;

Lm44: for A, B being Element of DEDEKIND_CUTS st A = { r where r is Element of RAT+ : r < one } holds
A *' B = B

proof end;

theorem :: ARYTM_2:16

for x, y being Element of REAL+ st x = one holds
x *' y = y

proof end;

Lm45: for i, j being Element of omega
for x9, y9 being Element of RAT+ st i = x9 & j = y9 holds
i +^ j = x9 + y9

proof end;

Lm46: for z9, x9, y9 being Element of RAT+ st z9 < x9 + y9 & x9 <> {} & y9 <> {} holds
ex r, s being Element of RAT+ st
( z9 = r + s & r < x9 & s < y9 )

proof end;

Lm47: for x, y being Element of REAL+ st x in RAT+ & y in RAT+ holds
ex x9, y9 being Element of RAT+ st
( x = x9 & y = y9 & x + y = x9 + y9 )

proof end;

theorem :: ARYTM_2:17

for x, y being Element of REAL+ st x in omega & y in omega holds
y + x in omega

proof end;

theorem :: ARYTM_2:18

for A being Subset of REAL+ st {} in A & ( for x, y being Element of REAL+ st x in A & y = one holds
x + y in A ) holds
omega c= A

proof end;

theorem :: ARYTM_2:19

for x being Element of REAL+ st x in omega holds
for y being Element of REAL+ holds
( y in x iff ( y in omega & y <> x & y <=' x ) )

proof end;

theorem :: ARYTM_2:20

for x, y, z being Element of REAL+ st x = y + z holds
z <=' x

proof end;

theorem :: ARYTM_2:21

( {} in REAL+ & one in REAL+ )

proof end;

theorem :: ARYTM_2:22

for x, y being Element of REAL+ st x in RAT+ & y in RAT+ holds
ex x9, y9 being Element of RAT+ st
( x = x9 & y = y9 & x *' y = x9 *' y9 )

proof end;