:: Semilattice Operations on Finite Subsets
:: by Andrzej Trybulec
::
:: Received September 18, 1989
:: Copyright (c) 1990-2011 Association of Mizar Users

begin

theorem :: SETWISEO:1

canceled;

theorem :: SETWISEO:2

canceled;

theorem Th3: :: SETWISEO:3

for x, y, z being set holds {x} c= {x,y,z}

proof end;

theorem Th4: :: SETWISEO:4

for x, y, z being set holds {x,y} c= {x,y,z}

proof end;

theorem Th5: :: SETWISEO:5

for X, Y, x being set holds
( not X c= Y \/ {x} or x in X or X c= Y )

proof end;

theorem Th6: :: SETWISEO:6

for x, X, y being set holds
( x in X \/ {y} iff ( x in X or x = y ) )

proof end;

theorem :: SETWISEO:7

canceled;

theorem Th8: :: SETWISEO:8

for X, x, Y being set holds
( X \/ {x} c= Y iff ( x in Y & X c= Y ) )

proof end;

theorem :: SETWISEO:9

canceled;

theorem :: SETWISEO:10

canceled;

theorem Th11: :: SETWISEO:11

for X, Y being set
for f being Function holds f .: (Y \ (f " X)) = (f .: Y) \ X

proof end;

theorem Th12: :: SETWISEO:12

for Y, X being non empty set
for f being Function of X,Y
for x being Element of X holds x in f " {(f . x)}

proof end;

theorem Th13: :: SETWISEO:13

for Y, X being non empty set
for f being Function of X,Y
for x being Element of X holds Im (f,x) = {(f . x)}

proof end;

theorem Th14: :: SETWISEO:14

for X being non empty set
for B being Element of Fin X
for x being set st x in B holds
x is Element of X

proof end;

theorem :: SETWISEO:15

for X, Y being non empty set
for A being Element of Fin X
for B being set
for f being Function of X,Y st ( for x being Element of X st x in A holds
f . x in B ) holds
f .: A c= B

proof end;

theorem Th16: :: SETWISEO:16

for X being set
for B being Element of Fin X
for A being set st A c= B holds
A is Element of Fin X

proof end;

Lm1: for X, Y being non empty set
for f being Function of X,Y
for A being Element of Fin X holds f .: A is Element of Fin Y
by FINSUB_1:def 5;

theorem :: SETWISEO:17

canceled;

theorem Th18: :: SETWISEO:18

for X being non empty set
for B being Element of Fin X st B <> {} holds
ex x being Element of X st x in B

proof end;

theorem Th19: :: SETWISEO:19

for Y, X being non empty set
for f being Function of X,Y
for A being Element of Fin X st f .: A = {} holds
A = {}

proof end;

registration

let X be set ;
cluster empty finite Element of Fin X;
existence
ex b₁ being Element of Fin X st b₁ is empty

proof end;

end;

definition

let X be set ;
func {}. X -> empty Element of Fin X equals :: SETWISEO:def 1
{} ;
coherence
{} is empty Element of Fin X by FINSUB_1:18;

end;

:: deftheorem defines {}. SETWISEO:def 1 :
for X being set holds {}. X = {} ;

scheme :: SETWISEO:sch 1
FinSubFuncEx{ F₁() -> non empty set , F₂() -> Element of Fin F₁(), P₁[ set , set ] } :

ex f being Function of F₁(),(Fin F₁()) st
for b, a being Element of F₁() holds
( a in f . b iff ( a in F₂() & P₁[a,b] ) )

proof end;

definition

let X be non empty set ;
let F be BinOp of X;
attr F is having_a_unity means :Def2: :: SETWISEO:def 2
ex x being Element of X st x is_a_unity_wrt F;

end;

:: deftheorem Def2 defines having_a_unity SETWISEO:def 2 :
for X being non empty set
for F being BinOp of X holds
( F is having_a_unity iff ex x being Element of X st x is_a_unity_wrt F );

theorem :: SETWISEO:20

canceled;

theorem :: SETWISEO:21

canceled;

theorem Th22: :: SETWISEO:22

for X being non empty set
for F being BinOp of X holds
( F is having_a_unity iff the_unity_wrt F is_a_unity_wrt F )

proof end;

theorem Th23: :: SETWISEO:23

for X being non empty set
for F being BinOp of X st F is having_a_unity holds
for x being Element of X holds
( F . ((the_unity_wrt F),x) = x & F . (x,(the_unity_wrt F)) = x )

proof end;

registration

let X be non empty set ;
cluster non empty finite Element of Fin X;
existence
not for b₁ being Element of Fin X holds b₁ is empty

proof end;

end;

notation

let X be non empty set ;
let x be Element of X;
synonym {.x.} for {X};
let y be Element of X;
synonym {.x,y.} for {X,x};
let z be Element of X;
synonym {.x,y,z.} for {X,x,y};

end;

definition

let X be non empty set ;
let x be Element of X;
:: original: {.
redefine func {.x.} -> Element of Fin X;
coherence
{..} is Element of Fin X

proof end;

let y be Element of X;
:: original: {.
redefine func {.x,y.} -> Element of Fin X;
coherence
{.y,.} is Element of Fin X

proof end;

let z be Element of X;
:: original: {.
redefine func {.x,y,z.} -> Element of Fin X;
coherence
{.y,z,.} is Element of Fin X

proof end;

end;

definition

let X be set ;
let A, B be Element of Fin X;
:: original: \/
redefine func A \/ B -> Element of Fin X;
coherence
A \/ B is Element of Fin X by FINSUB_1:10;

end;

definition

let X be set ;
let A, B be Element of Fin X;
:: original: \
redefine func A \ B -> Element of Fin X;
coherence
A \ B is Element of Fin X by FINSUB_1:10;

end;

scheme :: SETWISEO:sch 2
FinSubInd1{ F₁() -> non empty set , P₁[ set ] } :

for B being Element of Fin F₁() holds P₁[B]

provided

A1: P₁[ {}. F₁()] and
A2: for B9 being Element of Fin F₁()
for b being Element of F₁() st P₁[B9] & not b in B9 holds
P₁[B9 \/ {b}]

proof end;

scheme :: SETWISEO:sch 3
FinSubInd2{ F₁() -> non empty set , P₁[ Element of Fin F₁()] } :

for B being non empty Element of Fin F₁() holds P₁[B]

provided

A1: for x being Element of F₁() holds P₁[{.x.}] and
A2: for B1, B2 being non empty Element of Fin F₁() st P₁[B1] & P₁[B2] holds
P₁[B1 \/ B2]

proof end;

scheme :: SETWISEO:sch 4
FinSubInd3{ F₁() -> non empty set , P₁[ set ] } :

for B being Element of Fin F₁() holds P₁[B]

provided

A1: P₁[ {}. F₁()] and
A2: for B9 being Element of Fin F₁()
for b being Element of F₁() st P₁[B9] holds
P₁[B9 \/ {b}]

proof end;

definition

let X, Y be non empty set ;
let F be BinOp of Y;
let B be Element of Fin X;
let f be Function of X,Y;
assume that
A1: ( B <> {} or F is having_a_unity ) and
A2: F is commutative and
A3: F is associative ;
func F $$ (B,f) -> Element of Y means :Def3: :: SETWISEO:def 3
ex G being Function of (Fin X),Y st
( it = G . B & ( for e being Element of Y st e is_a_unity_wrt F holds
G . {} = e ) & ( for x being Element of X holds G . {x} = f . x ) & ( for B9 being Element of Fin X st B9 c= B & B9 <> {} holds
for x being Element of X st x in B \ B9 holds
G . (B9 \/ {x}) = F . ((G . B9),(f . x)) ) );
existence
ex b₁ being Element of Y ex G being Function of (Fin X),Y st
( b₁ = G . B & ( for e being Element of Y st e is_a_unity_wrt F holds
G . {} = e ) & ( for x being Element of X holds G . {x} = f . x ) & ( for B9 being Element of Fin X st B9 c= B & B9 <> {} holds
for x being Element of X st x in B \ B9 holds
G . (B9 \/ {x}) = F . ((G . B9),(f . x)) ) )

proof end;

uniqueness
for b₁, b₂ being Element of Y st ex G being Function of (Fin X),Y st
( b₁ = G . B & ( for e being Element of Y st e is_a_unity_wrt F holds
G . {} = e ) & ( for x being Element of X holds G . {x} = f . x ) & ( for B9 being Element of Fin X st B9 c= B & B9 <> {} holds
for x being Element of X st x in B \ B9 holds
G . (B9 \/ {x}) = F . ((G . B9),(f . x)) ) ) & ex G being Function of (Fin X),Y st
( b₂ = G . B & ( for e being Element of Y st e is_a_unity_wrt F holds
G . {} = e ) & ( for x being Element of X holds G . {x} = f . x ) & ( for B9 being Element of Fin X st B9 c= B & B9 <> {} holds
for x being Element of X st x in B \ B9 holds
G . (B9 \/ {x}) = F . ((G . B9),(f . x)) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def3 defines $$ SETWISEO:def 3 :
for X, Y being non empty set
for F being BinOp of Y
for B being Element of Fin X
for f being Function of X,Y st ( B <> {} or F is having_a_unity ) & F is commutative & F is associative holds
for b₆ being Element of Y holds
( b₆ = F $$ (B,f) iff ex G being Function of (Fin X),Y st
( b₆ = G . B & ( for e being Element of Y st e is_a_unity_wrt F holds
G . {} = e ) & ( for x being Element of X holds G . {x} = f . x ) & ( for B9 being Element of Fin X st B9 c= B & B9 <> {} holds
for x being Element of X st x in B \ B9 holds
G . (B9 \/ {x}) = F . ((G . B9),(f . x)) ) ) );

theorem :: SETWISEO:24

canceled;

theorem Th25: :: SETWISEO:25

for X, Y being non empty set
for F being BinOp of Y
for B being Element of Fin X
for f being Function of X,Y st ( B <> {} or F is having_a_unity ) & F is idempotent & F is commutative & F is associative holds
for IT being Element of Y holds
( IT = F $$ (B,f) iff ex G being Function of (Fin X),Y st
( IT = G . B & ( for e being Element of Y st e is_a_unity_wrt F holds
G . {} = e ) & ( for x being Element of X holds G . {x} = f . x ) & ( for B9 being Element of Fin X st B9 c= B & B9 <> {} holds
for x being Element of X st x in B holds
G . (B9 \/ {x}) = F . ((G . B9),(f . x)) ) ) )

proof end;

theorem Th26: :: SETWISEO:26

for Y, X being non empty set
for F being BinOp of Y
for f being Function of X,Y st F is commutative & F is associative holds
for b being Element of X holds F $$ ({.b.},f) = f . b

proof end;

theorem Th27: :: SETWISEO:27

for Y, X being non empty set
for F being BinOp of Y
for f being Function of X,Y st F is idempotent & F is commutative & F is associative holds
for a, b being Element of X holds F $$ ({.a,b.},f) = F . ((f . a),(f . b))

proof end;

theorem Th28: :: SETWISEO:28

for Y, X being non empty set
for F being BinOp of Y
for f being Function of X,Y st F is idempotent & F is commutative & F is associative holds
for a, b, c being Element of X holds F $$ ({.a,b,c.},f) = F . ((F . ((f . a),(f . b))),(f . c))

proof end;

theorem Th29: :: SETWISEO:29

for Y, X being non empty set
for F being BinOp of Y
for B being Element of Fin X
for f being Function of X,Y st F is idempotent & F is commutative & F is associative & B <> {} holds
for x being Element of X holds F $$ ((B \/ {.x.}),f) = F . ((F $$ (B,f)),(f . x))

proof end;

theorem Th30: :: SETWISEO:30

for Y, X being non empty set
for F being BinOp of Y
for f being Function of X,Y st F is idempotent & F is commutative & F is associative holds
for B1, B2 being Element of Fin X st B1 <> {} & B2 <> {} holds
F $$ ((B1 \/ B2),f) = F . ((F $$ (B1,f)),(F $$ (B2,f)))

proof end;

theorem :: SETWISEO:31

for Y, X being non empty set
for F being BinOp of Y
for B being Element of Fin X
for f being Function of X,Y st F is commutative & F is associative & F is idempotent holds
for x being Element of X st x in B holds
F . ((f . x),(F $$ (B,f))) = F $$ (B,f)

proof end;

theorem :: SETWISEO:32

for Y, X being non empty set
for F being BinOp of Y
for f being Function of X,Y st F is commutative & F is associative & F is idempotent holds
for B, C being Element of Fin X st B <> {} & B c= C holds
F . ((F $$ (B,f)),(F $$ (C,f))) = F $$ (C,f)

proof end;

theorem Th33: :: SETWISEO:33

for Y, X being non empty set
for F being BinOp of Y
for B being Element of Fin X
for f being Function of X,Y st B <> {} & F is commutative & F is associative & F is idempotent holds
for a being Element of Y st ( for b being Element of X st b in B holds
f . b = a ) holds
F $$ (B,f) = a

proof end;

theorem Th34: :: SETWISEO:34

for X, Y being non empty set
for F being BinOp of Y
for B being Element of Fin X
for f being Function of X,Y st F is commutative & F is associative & F is idempotent holds
for a being Element of Y st f .: B = {a} holds
F $$ (B,f) = a

proof end;

theorem Th35: :: SETWISEO:35

for X, Y being non empty set
for F being BinOp of Y st F is commutative & F is associative & F is idempotent holds
for f, g being Function of X,Y
for A, B being Element of Fin X st A <> {} & f .: A = g .: B holds
F $$ (A,f) = F $$ (B,g)

proof end;

theorem :: SETWISEO:36

for Y, X being non empty set
for F, G being BinOp of Y st F is idempotent & F is commutative & F is associative & G is_distributive_wrt F holds
for B being Element of Fin X st B <> {} holds
for f being Function of X,Y
for a being Element of Y holds G . (a,(F $$ (B,f))) = F $$ (B,(G [;] (a,f)))

proof end;

theorem :: SETWISEO:37

for Y, X being non empty set
for F, G being BinOp of Y st F is idempotent & F is commutative & F is associative & G is_distributive_wrt F holds
for B being Element of Fin X st B <> {} holds
for f being Function of X,Y
for a being Element of Y holds G . ((F $$ (B,f)),a) = F $$ (B,(G [:] (f,a)))

proof end;

definition

let X, Y be non empty set ;
let f be Function of X,Y;
let A be Element of Fin X;
:: original: .:
redefine func f .: A -> Element of Fin Y;
coherence
f .: A is Element of Fin Y by Lm1;

end;

theorem Th38: :: SETWISEO:38

for A, X, Y being non empty set
for F being BinOp of A st F is idempotent & F is commutative & F is associative holds
for B being Element of Fin X st B <> {} holds
for f being Function of X,Y
for g being Function of Y,A holds F $$ ((f .: B),g) = F $$ (B,(g * f))

proof end;

theorem Th39: :: SETWISEO:39

for X, Y being non empty set
for F being BinOp of Y st F is commutative & F is associative & F is idempotent holds
for Z being non empty set
for G being BinOp of Z st G is commutative & G is associative & G is idempotent holds
for f being Function of X,Y
for g being Function of Y,Z st ( for x, y being Element of Y holds g . (F . (x,y)) = G . ((g . x),(g . y)) ) holds
for B being Element of Fin X st B <> {} holds
g . (F $$ (B,f)) = G $$ (B,(g * f))

proof end;

theorem Th40: :: SETWISEO:40

for Y, X being non empty set
for F being BinOp of Y st F is commutative & F is associative & F is having_a_unity holds
for f being Function of X,Y holds F $$ (({}. X),f) = the_unity_wrt F

proof end;

theorem Th41: :: SETWISEO:41

for Y, X being non empty set
for F being BinOp of Y
for B being Element of Fin X
for f being Function of X,Y st F is idempotent & F is commutative & F is associative & F is having_a_unity holds
for x being Element of X holds F $$ ((B \/ {.x.}),f) = F . ((F $$ (B,f)),(f . x))

proof end;

theorem :: SETWISEO:42

for Y, X being non empty set
for F being BinOp of Y
for f being Function of X,Y st F is idempotent & F is commutative & F is associative & F is having_a_unity holds
for B1, B2 being Element of Fin X holds F $$ ((B1 \/ B2),f) = F . ((F $$ (B1,f)),(F $$ (B2,f)))

proof end;

theorem :: SETWISEO:43

for X, Y being non empty set
for F being BinOp of Y st F is commutative & F is associative & F is idempotent & F is having_a_unity holds
for f, g being Function of X,Y
for A, B being Element of Fin X st f .: A = g .: B holds
F $$ (A,f) = F $$ (B,g)

proof end;

theorem Th44: :: SETWISEO:44

for A, X, Y being non empty set
for F being BinOp of A st F is idempotent & F is commutative & F is associative & F is having_a_unity holds
for B being Element of Fin X
for f being Function of X,Y
for g being Function of Y,A holds F $$ ((f .: B),g) = F $$ (B,(g * f))

proof end;

theorem :: SETWISEO:45

for X, Y being non empty set
for F being BinOp of Y st F is commutative & F is associative & F is idempotent & F is having_a_unity holds
for Z being non empty set
for G being BinOp of Z st G is commutative & G is associative & G is idempotent & G is having_a_unity holds
for f being Function of X,Y
for g being Function of Y,Z st g . (the_unity_wrt F) = the_unity_wrt G & ( for x, y being Element of Y holds g . (F . (x,y)) = G . ((g . x),(g . y)) ) holds
for B being Element of Fin X holds g . (F $$ (B,f)) = G $$ (B,(g * f))

proof end;

definition

let A be set ;
func FinUnion A -> BinOp of (Fin A) means :Def4: :: SETWISEO:def 4
for x, y being Element of Fin A holds it . (x,y) = x \/ y;
existence
ex b₁ being BinOp of (Fin A) st
for x, y being Element of Fin A holds b₁ . (x,y) = x \/ y

proof end;

uniqueness
for b₁, b₂ being BinOp of (Fin A) st ( for x, y being Element of Fin A holds b₁ . (x,y) = x \/ y ) & ( for x, y being Element of Fin A holds b₂ . (x,y) = x \/ y ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def4 defines FinUnion SETWISEO:def 4 :
for A being set
for b₂ being BinOp of (Fin A) holds
( b₂ = FinUnion A iff for x, y being Element of Fin A holds b₂ . (x,y) = x \/ y );

theorem :: SETWISEO:46

canceled;

theorem :: SETWISEO:47

canceled;

theorem :: SETWISEO:48

canceled;

theorem Th49: :: SETWISEO:49

for A being set holds FinUnion A is idempotent

proof end;

theorem Th50: :: SETWISEO:50

for A being set holds FinUnion A is commutative

proof end;

theorem Th51: :: SETWISEO:51

for A being set holds FinUnion A is associative

proof end;

theorem Th52: :: SETWISEO:52

for A being set holds {}. A is_a_unity_wrt FinUnion A

proof end;

theorem Th53: :: SETWISEO:53

for A being set holds FinUnion A is having_a_unity

proof end;

theorem :: SETWISEO:54

for A being set holds the_unity_wrt (FinUnion A) is_a_unity_wrt FinUnion A

proof end;

theorem Th55: :: SETWISEO:55

for A being set holds the_unity_wrt (FinUnion A) = {}

proof end;

definition

let X be non empty set ;
let A be set ;
let B be Element of Fin X;
let f be Function of X,(Fin A);
func FinUnion (B,f) -> Element of Fin A equals :: SETWISEO:def 5
(FinUnion A) $$ (B,f);
coherence
(FinUnion A) $$ (B,f) is Element of Fin A ;

end;

:: deftheorem defines FinUnion SETWISEO:def 5 :
for X being non empty set
for A being set
for B being Element of Fin X
for f being Function of X,(Fin A) holds FinUnion (B,f) = (FinUnion A) $$ (B,f);

theorem :: SETWISEO:56

for X being non empty set
for A being set
for f being Function of X,(Fin A)
for i being Element of X holds FinUnion ({.i.},f) = f . i

proof end;

theorem :: SETWISEO:57

for X being non empty set
for A being set
for f being Function of X,(Fin A)
for i, j being Element of X holds FinUnion ({.i,j.},f) = (f . i) \/ (f . j)

proof end;

theorem :: SETWISEO:58

for X being non empty set
for A being set
for f being Function of X,(Fin A)
for i, j, k being Element of X holds FinUnion ({.i,j,k.},f) = ((f . i) \/ (f . j)) \/ (f . k)

proof end;

theorem Th59: :: SETWISEO:59

for X being non empty set
for A being set
for f being Function of X,(Fin A) holds FinUnion (({}. X),f) = {}

proof end;

theorem Th60: :: SETWISEO:60

for X being non empty set
for A being set
for f being Function of X,(Fin A)
for i being Element of X
for B being Element of Fin X holds FinUnion ((B \/ {.i.}),f) = (FinUnion (B,f)) \/ (f . i)

proof end;

theorem Th61: :: SETWISEO:61

for X being non empty set
for A being set
for f being Function of X,(Fin A)
for B being Element of Fin X holds FinUnion (B,f) = union (f .: B)

proof end;

theorem :: SETWISEO:62

for X being non empty set
for A being set
for f being Function of X,(Fin A)
for B1, B2 being Element of Fin X holds FinUnion ((B1 \/ B2),f) = (FinUnion (B1,f)) \/ (FinUnion (B2,f))

proof end;

theorem :: SETWISEO:63

for X, Y being non empty set
for A being set
for B being Element of Fin X
for f being Function of X,Y
for g being Function of Y,(Fin A) holds FinUnion ((f .: B),g) = FinUnion (B,(g * f))

proof end;

theorem Th64: :: SETWISEO:64

for A, X being non empty set
for Y being set
for G being BinOp of A st G is commutative & G is associative & G is idempotent holds
for B being Element of Fin X st B <> {} holds
for f being Function of X,(Fin Y)
for g being Function of (Fin Y),A st ( for x, y being Element of Fin Y holds g . (x \/ y) = G . ((g . x),(g . y)) ) holds
g . (FinUnion (B,f)) = G $$ (B,(g * f))

proof end;

theorem Th65: :: SETWISEO:65

for X, Z being non empty set
for Y being set
for G being BinOp of Z st G is commutative & G is associative & G is idempotent & G is having_a_unity holds
for f being Function of X,(Fin Y)
for g being Function of (Fin Y),Z st g . ({}. Y) = the_unity_wrt G & ( for x, y being Element of Fin Y holds g . (x \/ y) = G . ((g . x),(g . y)) ) holds
for B being Element of Fin X holds g . (FinUnion (B,f)) = G $$ (B,(g * f))

proof end;

definition

let A be set ;
func singleton A -> Function of A,(Fin A) means :Def6: :: SETWISEO:def 6
for x being set st x in A holds
it . x = {x};
existence
ex b₁ being Function of A,(Fin A) st
for x being set st x in A holds
b₁ . x = {x}

proof end;

uniqueness
for b₁, b₂ being Function of A,(Fin A) st ( for x being set st x in A holds
b₁ . x = {x} ) & ( for x being set st x in A holds
b₂ . x = {x} ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def6 defines singleton SETWISEO:def 6 :
for A being set
for b₂ being Function of A,(Fin A) holds
( b₂ = singleton A iff for x being set st x in A holds
b₂ . x = {x} );

theorem :: SETWISEO:66

canceled;

theorem Th67: :: SETWISEO:67

for A being non empty set
for f being Function of A,(Fin A) holds
( f = singleton A iff for x being Element of A holds f . x = {x} )

proof end;

theorem Th68: :: SETWISEO:68

for X being non empty set
for x being set
for y being Element of X holds
( x in (singleton X) . y iff x = y )

proof end;

theorem :: SETWISEO:69

for X being non empty set
for x, y, z being Element of X st x in (singleton X) . z & y in (singleton X) . z holds
x = y

proof end;

Lm2: for D being non empty set
for X, P being set
for f being Function of X,D holds f .: P c= D
;

theorem Th70: :: SETWISEO:70

for X being non empty set
for A being set
for f being Function of X,(Fin A)
for B being Element of Fin X
for x being set holds
( x in FinUnion (B,f) iff ex i being Element of X st
( i in B & x in f . i ) )

proof end;

theorem :: SETWISEO:71

for X being non empty set
for B being Element of Fin X holds FinUnion (B,(singleton X)) = B

proof end;

theorem :: SETWISEO:72

for X being non empty set
for Y, Z being set
for f being Function of X,(Fin Y)
for g being Function of (Fin Y),(Fin Z) st g . ({}. Y) = {}. Z & ( for x, y being Element of Fin Y holds g . (x \/ y) = (g . x) \/ (g . y) ) holds
for B being Element of Fin X holds g . (FinUnion (B,f)) = FinUnion (B,(g * f))

proof end;