:: Partially Ordered Sets
:: by Wojciech A. Trybulec
::
:: Received August 30, 1989
:: Copyright (c) 1990-2021 Association of Mizar Users

Lm1: for Y being set holds
( ex X being set st
( X <> {} & X in Y ) iff union Y <> {} )

proof end;

::
:: Choice function.
::

definition

let f be Function;

mode Choice of f -> Function means :Def1: :: ORDERS_1:def 1
( dom it = dom f & ( for x being object st x in dom f holds
it . x = the Element of f . x ) );

existence
ex b₁ being Function st
( dom b₁ = dom f & ( for x being object st x in dom f holds
b₁ . x = the Element of f . x ) )

proof end;

end;

:: deftheorem Def1 defines Choice ORDERS_1:def 1 :
for f, b₂ being Function holds
( b₂ is Choice of f iff ( dom b₂ = dom f & ( for x being object st x in dom f holds
b₂ . x = the Element of f . x ) ) );

definition

let I be set ;
let M be ManySortedSet of I;

:: original: Choice
redefine mode Choice of M -> ManySortedSet of I means :Def2: :: ORDERS_1:def 2
for x being object st x in I holds
it . x = the Element of M . x;

coherence
for b₁ being Choice of M holds b₁ is ManySortedSet of I

proof end;

compatibility
for b₁ being ManySortedSet of I holds
( b₁ is Choice of M iff for x being object st x in I holds
b₁ . x = the Element of M . x )

proof end;

end;

:: deftheorem Def2 defines Choice ORDERS_1:def 2 :
for I being set
for M, b₃ being ManySortedSet of I holds
( b₃ is Choice of M iff for x being object st x in I holds
b₃ . x = the Element of M . x );

definition

let A be set ;
mode Choice_Function of A is Choice of id A;

end;

Lm2: for M being non empty set st not {} in M holds
for Ch being Choice_Function of M
for X being set st X in M holds
Ch . X in X

proof end;

Lm3: for M being non empty set st not {} in M holds
for Ch being Choice_Function of M holds Ch is Function of M,(union M)

proof end;

definition

let D be set ;

func BOOL D -> set equals :: ORDERS_1:def 3
(bool D) \ {{}};

coherence
(bool D) \ {{}} is set ;

end;

:: deftheorem defines BOOL ORDERS_1:def 3 :
for D being set holds BOOL D = (bool D) \ {{}};

registration

let D be non empty set ;

cluster BOOL D -> non empty ;

coherence
not BOOL D is empty

proof end;

end;

theorem :: ORDERS_1:1

for D being non empty set holds not {} in BOOL D

proof end;

theorem :: ORDERS_1:2

for X being set
for D being non empty set holds
( D c= X iff D in BOOL X )

proof end;

::
:: Orders.
::

definition

let X be set ;
mode Order of X is reflexive antisymmetric transitive total Relation of X;

end;

Lm4: for X being set
for R being total Relation of X holds field R = X

proof end;

theorem Th3: :: ORDERS_1:3

for X, x being set
for O being Order of X st x in X holds
[x,x] in O

proof end;

theorem :: ORDERS_1:4

for X, x, y being set
for O being Order of X st x in X & y in X & [x,y] in O & [y,x] in O holds
x = y

proof end;

theorem :: ORDERS_1:5

for X, x, y, z being set
for O being Order of X st x in X & y in X & z in X & [x,y] in O & [y,z] in O holds
[x,z] in O

proof end;

theorem :: ORDERS_1:6

for Y being set holds
( ex X being set st
( X <> {} & X in Y ) iff union Y <> {} ) by Lm1;

theorem :: ORDERS_1:7

for X being set
for P being Relation holds
( P is_strongly_connected_in X iff ( P is_reflexive_in X & P is_connected_in X ) )

proof end;

theorem :: ORDERS_1:8

for X, Y being set
for P being Relation st P is_reflexive_in X & Y c= X holds
P is_reflexive_in Y ;

theorem :: ORDERS_1:9

for X, Y being set
for P being Relation st P is_antisymmetric_in X & Y c= X holds
P is_antisymmetric_in Y ;

theorem :: ORDERS_1:10

for X, Y being set
for P being Relation st P is_transitive_in X & Y c= X holds
P is_transitive_in Y ;

theorem :: ORDERS_1:11

for X, Y being set
for P being Relation st P is_strongly_connected_in X & Y c= X holds
P is_strongly_connected_in Y ;

theorem :: ORDERS_1:12

for X being set
for R being total Relation of X holds field R = X by Lm4;

theorem Th13: :: ORDERS_1:13

for A being set
for R being Relation of A st R is_reflexive_in A holds
( dom R = A & field R = A )

proof end;

::
:: Orders.
::

theorem Th14: :: ORDERS_1:14

for X being set
for O being Order of X holds
( dom O = X & rng O = X )

proof end;

theorem :: ORDERS_1:15

for X being set
for O being Order of X holds field O = X

proof end;

definition

let R be Relation;

attr R is being_quasi-order means :: ORDERS_1:def 4
( R is reflexive & R is transitive );

attr R is being_partial-order means :: ORDERS_1:def 5
( R is reflexive & R is transitive & R is antisymmetric );

attr R is being_linear-order means :: ORDERS_1:def 6
( R is reflexive & R is transitive & R is antisymmetric & R is connected );

end;

:: deftheorem defines being_quasi-order ORDERS_1:def 4 :
for R being Relation holds
( R is being_quasi-order iff ( R is reflexive & R is transitive ) );

:: deftheorem defines being_partial-order ORDERS_1:def 5 :
for R being Relation holds
( R is being_partial-order iff ( R is reflexive & R is transitive & R is antisymmetric ) );

:: deftheorem defines being_linear-order ORDERS_1:def 6 :
for R being Relation holds
( R is being_linear-order iff ( R is reflexive & R is transitive & R is antisymmetric & R is connected ) );

theorem :: ORDERS_1:16

for R being Relation st R is being_quasi-order holds
R ~ is being_quasi-order ;

theorem :: ORDERS_1:17

for R being Relation st R is being_partial-order holds
R ~ is being_partial-order ;

Lm5: for R being Relation st R is connected holds
R ~ is connected

proof end;

theorem Th18: :: ORDERS_1:18

for R being Relation st R is being_linear-order holds
R ~ is being_linear-order by Lm5;

theorem :: ORDERS_1:19

for R being Relation st R is well-ordering holds
( R is being_quasi-order & R is being_partial-order & R is being_linear-order ) ;

theorem :: ORDERS_1:20

for R being Relation st R is being_linear-order holds
( R is being_quasi-order & R is being_partial-order ) ;

theorem :: ORDERS_1:21

for R being Relation st R is being_partial-order holds
R is being_quasi-order ;

theorem :: ORDERS_1:22

for X being set
for O being Order of X holds O is being_partial-order ;

theorem :: ORDERS_1:23

for X being set
for O being Order of X holds O is being_quasi-order ;

theorem :: ORDERS_1:24

for X being set
for O being Order of X st O is connected holds
O is being_linear-order ;

Lm6: for R being Relation holds R c= [:(field R),(field R):]

proof end;

Lm7: for R being Relation
for X being set st R is reflexive & X c= field R holds
field (R |_2 X) = X

proof end;

theorem :: ORDERS_1:25

for R being Relation
for X being set st R is being_quasi-order holds
R |_2 X is being_quasi-order by WELLORD1:15, WELLORD1:17;

theorem :: ORDERS_1:26

for R being Relation
for X being set st R is being_partial-order holds
R |_2 X is being_partial-order by WELLORD1:15, WELLORD1:17;

theorem :: ORDERS_1:27

for R being Relation
for X being set st R is being_linear-order holds
R |_2 X is being_linear-order by WELLORD1:15, WELLORD1:16, WELLORD1:17;

registration

let R be empty Relation;

cluster field R -> empty ;

coherence
field R is empty ;

end;

registration

cluster empty Relation-like -> well-ordering being_quasi-order being_partial-order being_linear-order for set ;

coherence
for b₁ being Relation st b₁ is empty holds
( b₁ is being_quasi-order & b₁ is being_partial-order & b₁ is being_linear-order & b₁ is well-ordering )

proof end;

end;

registration

let X be set ;

cluster id X -> being_quasi-order being_partial-order ;

coherence
( id X is being_quasi-order & id X is being_partial-order ) ;

end;

definition

let R be Relation;
let X be set ;

pred R quasi_orders X means :: ORDERS_1:def 7
( R is_reflexive_in X & R is_transitive_in X );

pred R partially_orders X means :: ORDERS_1:def 8
( R is_reflexive_in X & R is_transitive_in X & R is_antisymmetric_in X );

pred R linearly_orders X means :: ORDERS_1:def 9
( R is_reflexive_in X & R is_transitive_in X & R is_antisymmetric_in X & R is_connected_in X );

end;

:: deftheorem defines quasi_orders ORDERS_1:def 7 :
for R being Relation
for X being set holds
( R quasi_orders X iff ( R is_reflexive_in X & R is_transitive_in X ) );

:: deftheorem defines partially_orders ORDERS_1:def 8 :
for R being Relation
for X being set holds
( R partially_orders X iff ( R is_reflexive_in X & R is_transitive_in X & R is_antisymmetric_in X ) );

:: deftheorem defines linearly_orders ORDERS_1:def 9 :
for R being Relation
for X being set holds
( R linearly_orders X iff ( R is_reflexive_in X & R is_transitive_in X & R is_antisymmetric_in X & R is_connected_in X ) );

theorem Th28: :: ORDERS_1:28

for R being Relation
for X being set st R well_orders X holds
( R quasi_orders X & R partially_orders X & R linearly_orders X ) ;

theorem Th29: :: ORDERS_1:29

for R being Relation
for X being set st R linearly_orders X holds
( R quasi_orders X & R partially_orders X ) ;

theorem :: ORDERS_1:30

for R being Relation
for X being set st R partially_orders X holds
R quasi_orders X ;

theorem Th31: :: ORDERS_1:31

for R being Relation st R is being_quasi-order holds
R quasi_orders field R

proof end;

theorem :: ORDERS_1:32

for R being Relation
for X, Y being set st R quasi_orders Y & X c= Y holds
R quasi_orders X

proof end;

Lm8: for R being Relation
for X being set st R is_reflexive_in X holds
R |_2 X is reflexive

proof end;

Lm9: for R being Relation
for X being set st R is_transitive_in X holds
R |_2 X is transitive

proof end;

Lm10: for R being Relation
for X being set st R is_antisymmetric_in X holds
R |_2 X is antisymmetric

proof end;

Lm11: for R being Relation
for X being set st R is_connected_in X holds
R |_2 X is connected

proof end;

theorem :: ORDERS_1:33

for R being Relation
for X being set st R quasi_orders X holds
R |_2 X is being_quasi-order by Lm8, Lm9;

theorem Th34: :: ORDERS_1:34

for R being Relation st R is being_partial-order holds
R partially_orders field R

proof end;

theorem :: ORDERS_1:35

for R being Relation
for X, Y being set st R partially_orders Y & X c= Y holds
R partially_orders X

proof end;

theorem :: ORDERS_1:36

for R being Relation
for X being set st R partially_orders X holds
R |_2 X is being_partial-order by Lm8, Lm9, Lm10;

theorem :: ORDERS_1:37

for R being Relation st R is being_linear-order holds
R linearly_orders field R

proof end;

theorem :: ORDERS_1:38

for R being Relation
for X, Y being set st R linearly_orders Y & X c= Y holds
R linearly_orders X

proof end;

theorem :: ORDERS_1:39

for R being Relation
for X being set st R linearly_orders X holds
R |_2 X is being_linear-order by Lm8, Lm9, Lm10, Lm11;

Lm12: for R being Relation
for X being set st R is_reflexive_in X holds
R ~ is_reflexive_in X

proof end;

Lm13: for R being Relation
for X being set st R is_transitive_in X holds
R ~ is_transitive_in X

proof end;

Lm14: for R being Relation
for X being set st R is_antisymmetric_in X holds
R ~ is_antisymmetric_in X

proof end;

Lm15: for R being Relation
for X being set st R is_connected_in X holds
R ~ is_connected_in X

proof end;

theorem :: ORDERS_1:40

for R being Relation
for X being set st R quasi_orders X holds
R ~ quasi_orders X by Lm12, Lm13;

theorem Th41: :: ORDERS_1:41

for R being Relation
for X being set st R partially_orders X holds
R ~ partially_orders X by Lm12, Lm13, Lm14;

theorem :: ORDERS_1:42

for R being Relation
for X being set st R linearly_orders X holds
R ~ linearly_orders X by Lm12, Lm13, Lm14, Lm15;

theorem :: ORDERS_1:43

for X being set
for O being Order of X holds O quasi_orders X

proof end;

theorem :: ORDERS_1:44

for X being set
for O being Order of X holds O partially_orders X

proof end;

theorem Th45: :: ORDERS_1:45

for R being Relation
for X being set st R partially_orders X holds
R |_2 X is Order of X

proof end;

theorem :: ORDERS_1:46

for R being Relation
for X being set st R linearly_orders X holds
R |_2 X is Order of X by Th29, Th45;

theorem :: ORDERS_1:47

for R being Relation
for X being set st R well_orders X holds
R |_2 X is Order of X by Th28, Th45;

theorem :: ORDERS_1:48

for X being set holds
( id X quasi_orders X & id X partially_orders X )

proof end;

definition

let R be Relation;
let X be set ;

pred X has_upper_Zorn_property_wrt R means :: ORDERS_1:def 10
for Y being set st Y c= X & R |_2 Y is being_linear-order holds
ex x being set st
( x in X & ( for y being set st y in Y holds
[y,x] in R ) );

pred X has_lower_Zorn_property_wrt R means :: ORDERS_1:def 11
for Y being set st Y c= X & R |_2 Y is being_linear-order holds
ex x being set st
( x in X & ( for y being set st y in Y holds
[x,y] in R ) );

end;

:: deftheorem defines has_upper_Zorn_property_wrt ORDERS_1:def 10 :
for R being Relation
for X being set holds
( X has_upper_Zorn_property_wrt R iff for Y being set st Y c= X & R |_2 Y is being_linear-order holds
ex x being set st
( x in X & ( for y being set st y in Y holds
[y,x] in R ) ) );

:: deftheorem defines has_lower_Zorn_property_wrt ORDERS_1:def 11 :
for R being Relation
for X being set holds
( X has_lower_Zorn_property_wrt R iff for Y being set st Y c= X & R |_2 Y is being_linear-order holds
ex x being set st
( x in X & ( for y being set st y in Y holds
[x,y] in R ) ) );

Lm16: for R being Relation
for X being set holds (R |_2 X) ~ = (R ~) |_2 X

proof end;

Lm17: now :: thesis: for R being Relation holds R |_2 {} = {}

let R be Relation; :: thesis: R |_2 {} = {}
thus R |_2 {} = ({} |` R) | {} by WELLORD1:10
.= {} ; :: thesis: verum

end;

theorem Th49: :: ORDERS_1:49

for R being Relation
for X being set st X has_upper_Zorn_property_wrt R holds
X <> {}

proof end;

theorem :: ORDERS_1:50

for R being Relation
for X being set st X has_lower_Zorn_property_wrt R holds
X <> {}

proof end;

theorem Th51: :: ORDERS_1:51

for R being Relation
for X being set holds
( X has_upper_Zorn_property_wrt R iff X has_lower_Zorn_property_wrt R ~ )

proof end;

theorem :: ORDERS_1:52

for R being Relation
for X being set holds
( X has_upper_Zorn_property_wrt R ~ iff X has_lower_Zorn_property_wrt R )

proof end;

definition

let R be Relation;
let x be set ;

pred x is_maximal_in R means :: ORDERS_1:def 12
( x in field R & ( for y being set holds
( not y in field R or not y <> x or not [x,y] in R ) ) );

pred x is_minimal_in R means :: ORDERS_1:def 13
( x in field R & ( for y being set holds
( not y in field R or not y <> x or not [y,x] in R ) ) );

pred x is_superior_of R means :: ORDERS_1:def 14
( x in field R & ( for y being set st y in field R & y <> x holds
[y,x] in R ) );

pred x is_inferior_of R means :: ORDERS_1:def 15
( x in field R & ( for y being set st y in field R & y <> x holds
[x,y] in R ) );

end;

:: deftheorem defines is_maximal_in ORDERS_1:def 12 :
for R being Relation
for x being set holds
( x is_maximal_in R iff ( x in field R & ( for y being set holds
( not y in field R or not y <> x or not [x,y] in R ) ) ) );

:: deftheorem defines is_minimal_in ORDERS_1:def 13 :
for R being Relation
for x being set holds
( x is_minimal_in R iff ( x in field R & ( for y being set holds
( not y in field R or not y <> x or not [y,x] in R ) ) ) );

:: deftheorem defines is_superior_of ORDERS_1:def 14 :
for R being Relation
for x being set holds
( x is_superior_of R iff ( x in field R & ( for y being set st y in field R & y <> x holds
[y,x] in R ) ) );

:: deftheorem defines is_inferior_of ORDERS_1:def 15 :
for R being Relation
for x being set holds
( x is_inferior_of R iff ( x in field R & ( for y being set st y in field R & y <> x holds
[x,y] in R ) ) );

theorem :: ORDERS_1:53

for R being Relation
for x being set st x is_inferior_of R & R is antisymmetric holds
x is_minimal_in R

proof end;

theorem :: ORDERS_1:54

for R being Relation
for x being set st x is_superior_of R & R is antisymmetric holds
x is_maximal_in R

proof end;

theorem :: ORDERS_1:55

for R being Relation
for x being set st x is_minimal_in R & R is connected holds
x is_inferior_of R

proof end;

theorem :: ORDERS_1:56

for R being Relation
for x being set st x is_maximal_in R & R is connected holds
x is_superior_of R

proof end;

theorem :: ORDERS_1:57

for R being Relation
for X, x being set st x in X & x is_superior_of R & X c= field R & R is reflexive holds
X has_upper_Zorn_property_wrt R

proof end;

theorem :: ORDERS_1:58

for R being Relation
for X, x being set st x in X & x is_inferior_of R & X c= field R & R is reflexive holds
X has_lower_Zorn_property_wrt R

proof end;

theorem Th59: :: ORDERS_1:59

for R being Relation
for x being set holds
( x is_minimal_in R iff x is_maximal_in R ~ )

proof end;

theorem :: ORDERS_1:60

for R being Relation
for x being set holds
( x is_minimal_in R ~ iff x is_maximal_in R )

proof end;

theorem :: ORDERS_1:61

for R being Relation
for x being set holds
( x is_inferior_of R iff x is_superior_of R ~ )

proof end;

theorem :: ORDERS_1:62

for R being Relation
for x being set holds
( x is_inferior_of R ~ iff x is_superior_of R )

proof end;

Lm18: for R being Relation
for X, Y being set st R well_orders X & Y c= X holds
R well_orders Y

proof end;

theorem Th63: :: ORDERS_1:63

for R being Relation
for X being set st R partially_orders X & field R = X & X has_upper_Zorn_property_wrt R holds
ex x being set st x is_maximal_in R

proof end;

theorem Th64: :: ORDERS_1:64

for R being Relation
for X being set st R partially_orders X & field R = X & X has_lower_Zorn_property_wrt R holds
ex x being set st x is_minimal_in R

proof end;

:: Kuratowski-Zorn Lemma
:: Zorn Lemma

theorem Th65: :: ORDERS_1:65

for X being set st X <> {} & ( for Z being set st Z c= X & Z is c=-linear holds
ex Y being set st
( Y in X & ( for X1 being set st X1 in Z holds
X1 c= Y ) ) ) holds
ex Y being set st
( Y in X & ( for Z being set st Z in X & Z <> Y holds
not Y c= Z ) )

proof end;

theorem Th66: :: ORDERS_1:66

for X being set st X <> {} & ( for Z being set st Z c= X & Z is c=-linear holds
ex Y being set st
( Y in X & ( for X1 being set st X1 in Z holds
Y c= X1 ) ) ) holds
ex Y being set st
( Y in X & ( for Z being set st Z in X & Z <> Y holds
not Z c= Y ) )

proof end;

theorem Th67: :: ORDERS_1:67

for X being set st X <> {} & ( for Z being set st Z <> {} & Z c= X & Z is c=-linear holds
union Z in X ) holds
ex Y being set st
( Y in X & ( for Z being set st Z in X & Z <> Y holds
not Y c= Z ) )

proof end;

theorem :: ORDERS_1:68

for X being set st X <> {} & ( for Z being set st Z <> {} & Z c= X & Z is c=-linear holds
meet Z in X ) holds
ex Y being set st
( Y in X & ( for Z being set st Z in X & Z <> Y holds
not Z c= Y ) )

proof end;

scheme :: ORDERS_1:sch 1
ZornMax{ F₁() -> non empty set , P₁[ set , set ] } :

ex x being Element of F₁() st
for y being Element of F₁() st x <> y holds
not P₁[x,y]

provided

A1: for x being Element of F₁() holds P₁[x,x] and
A2: for x, y being Element of F₁() st P₁[x,y] & P₁[y,x] holds
x = y and
A3: for x, y, z being Element of F₁() st P₁[x,y] & P₁[y,z] holds
P₁[x,z] and
A4: for X being set st X c= F₁() & ( for x, y being Element of F₁() st x in X & y in X & P₁[x,y] holds
P₁[y,x] ) holds
ex y being Element of F₁() st
for x being Element of F₁() st x in X holds
P₁[x,y]

proof end;

scheme :: ORDERS_1:sch 2
ZornMin{ F₁() -> non empty set , P₁[ set , set ] } :

ex x being Element of F₁() st
for y being Element of F₁() st x <> y holds
not P₁[y,x]

provided

A1: for x being Element of F₁() holds P₁[x,x] and
A2: for x, y being Element of F₁() st P₁[x,y] & P₁[y,x] holds
x = y and
A3: for x, y, z being Element of F₁() st P₁[x,y] & P₁[y,z] holds
P₁[x,z] and
A4: for X being set st X c= F₁() & ( for x, y being Element of F₁() st x in X & y in X & P₁[x,y] holds
P₁[y,x] ) holds
ex y being Element of F₁() st
for x being Element of F₁() st x in X holds
P₁[y,x]

proof end;

::
:: Orders - continuation.
::

theorem :: ORDERS_1:69

for R being Relation
for X being set st R partially_orders X & field R = X holds
ex P being Relation st
( R c= P & P linearly_orders X & field P = X )

proof end;

::
:: Auxiliary theorems.
::

theorem :: ORDERS_1:70

for R being Relation holds R c= [:(field R),(field R):] by Lm6;

theorem :: ORDERS_1:71

for R being Relation
for X being set st R is reflexive & X c= field R holds
field (R |_2 X) = X by Lm7;

theorem :: ORDERS_1:72

for R being Relation
for X being set st R is_reflexive_in X holds
R |_2 X is reflexive by Lm8;

theorem :: ORDERS_1:73

for R being Relation
for X being set st R is_transitive_in X holds
R |_2 X is transitive by Lm9;

theorem :: ORDERS_1:74

for R being Relation
for X being set st R is_antisymmetric_in X holds
R |_2 X is antisymmetric by Lm10;

theorem :: ORDERS_1:75

for R being Relation
for X being set st R is_connected_in X holds
R |_2 X is connected by Lm11;

theorem :: ORDERS_1:76

for R being Relation
for X, Y being set st R is_connected_in X & Y c= X holds
R is_connected_in Y ;

theorem :: ORDERS_1:77

for R being Relation
for X, Y being set st R well_orders X & Y c= X holds
R well_orders Y by Lm18;

theorem :: ORDERS_1:78

for R being Relation st R is connected holds
R ~ is connected by Lm5;

theorem :: ORDERS_1:79

for R being Relation
for X being set st R is_reflexive_in X holds
R ~ is_reflexive_in X by Lm12;

theorem :: ORDERS_1:80

for R being Relation
for X being set st R is_transitive_in X holds
R ~ is_transitive_in X by Lm13;

theorem :: ORDERS_1:81

for R being Relation
for X being set st R is_antisymmetric_in X holds
R ~ is_antisymmetric_in X by Lm14;

theorem :: ORDERS_1:82

for R being Relation
for X being set st R is_connected_in X holds
R ~ is_connected_in X by Lm15;

theorem :: ORDERS_1:83

for R being Relation
for X being set holds (R |_2 X) ~ = (R ~) |_2 X by Lm16;

theorem :: ORDERS_1:84

for R being Relation holds R |_2 {} = {} by Lm17;

:: from COMPTS_1

theorem :: ORDERS_1:85

for f being Function
for Z being set st Z is finite & Z c= rng f holds
ex Y being set st
( Y c= dom f & Y is finite & f .: Y = Z )

proof end;

:: from AMISTD_3, 2006.03.26, A.T.

theorem Th86: :: ORDERS_1:86

for R being Relation st field R is finite holds
R is finite

proof end;

theorem :: ORDERS_1:87

for R being Relation st dom R is finite & rng R is finite holds
R is finite

proof end;

:: from AMISTD_3, 2010.01.10, A.T

registration

cluster order_type_of {} -> empty ;

coherence
order_type_of {} is empty

proof end;

end;

theorem :: ORDERS_1:88

for O being Ordinal holds order_type_of (RelIncl O) = O

proof end;

definition

let X be set ;
:: original: RelIncl
redefine func RelIncl X -> Order of X;
coherence
RelIncl X is Order of X

proof end;

end;

theorem :: ORDERS_1:89

for M being non empty set st not {} in M holds
for Ch being Choice_Function of M
for X being set st X in M holds
Ch . X in X by Lm2;

theorem :: ORDERS_1:90

for M being non empty set st not {} in M holds
for Ch being Choice_Function of M holds Ch is Function of M,(union M) by Lm3;