PBOOLE: Manysorted Sets

:: Manysorted Sets
:: by Andrzej Trybulec
::
:: Received July 7, 1993
:: Copyright (c) 1993-2012 Association of Mizar Users

begin

theorem :: PBOOLE:1

for f being Function st f is non-empty holds
rng f is with_non-empty_elements ;

theorem :: PBOOLE:2

for f being Function holds
( f is empty-yielding iff ( f = {} or rng f = {{}} ) )

proof end;

registration

let I be set ;

cluster Relation-like I -defined Function-like total for set ;

existence

proof end;

end;

definition

let I be set ;
mode ManySortedSet of I is I -defined total Function;

end;

scheme :: PBOOLE:sch 1
KuratowskiFunction{ F₁() -> set , F₂( set ) -> set } :

ex f being ManySortedSet of F₁() st
for e being set st e in F₁() holds
f . e in F₂(e)

provided

A1: for e being set st e in F₁() holds
F₂(e) <> {}

proof end;

definition

let I be set ;
let X, Y be ManySortedSet of I;

pred X in Y means :Def1: :: PBOOLE:def 1
for i being set st i in I holds
X . i in Y . i;

pred X c= Y means :Def2: :: PBOOLE:def 2
for i being set st i in I holds
X . i c= Y . i;

reflexivity ;

end;

:: deftheorem Def1 defines in PBOOLE:def 1 :

:: deftheorem Def2 defines c= PBOOLE:def 2 :

definition

let I be non empty set ;
let X, Y be ManySortedSet of I;
:: original: in
redefine pred X in Y;
asymmetry

proof end;

end;

scheme :: PBOOLE:sch 2
PSeparation{ F₁() -> set , F₂() -> ManySortedSet of F₁(), P₁[ set , set ] } :

ex X being ManySortedSet of F₁() st
for i being set st i in F₁() holds
for e being set holds
( e in X . i iff ( e in F₂() . i & P₁[i,e] ) )

proof end;

theorem Th3: :: PBOOLE:3

for I being set
for X, Y being ManySortedSet of I st ( for i being set st i in I holds
X . i = Y . i ) holds
X = Y

proof end;

definition

let I be set ;

func [[0]] I -> ManySortedSet of I equals :: PBOOLE:def 3
I --> {};

coherence ;
correctness ;
let X, Y be ManySortedSet of I;

func X \/ Y -> ManySortedSet of I means :Def4: :: PBOOLE:def 4
for i being set st i in I holds
it . i = (X . i) \/ (Y . i);

existence

proof end;

uniqueness

proof end;

commutativity ;
idempotence ;

func X /\ Y -> ManySortedSet of I means :Def5: :: PBOOLE:def 5
for i being set st i in I holds
it . i = (X . i) /\ (Y . i);

existence

proof end;

uniqueness

proof end;

commutativity ;
idempotence ;

func X \ Y -> ManySortedSet of I means :Def6: :: PBOOLE:def 6
for i being set st i in I holds
it . i = (X . i) \ (Y . i);

existence

proof end;

uniqueness

proof end;

pred X overlaps Y means :Def7: :: PBOOLE:def 7
for i being set st i in I holds
X . i meets Y . i;

symmetry ;

pred X misses Y means :Def8: :: PBOOLE:def 8
for i being set st i in I holds
X . i misses Y . i;

symmetry ;

end;

:: deftheorem defines [[0]] PBOOLE:def 3 :

:: deftheorem Def4 defines \/ PBOOLE:def 4 :

:: deftheorem Def5 defines /\ PBOOLE:def 5 :

:: deftheorem Def6 defines \ PBOOLE:def 6 :

:: deftheorem Def7 defines overlaps PBOOLE:def 7 :

:: deftheorem Def8 defines misses PBOOLE:def 8 :

notation

let I be set ;
let X, Y be ManySortedSet of I;
antonym X meets Y for X misses Y;

end;

definition

let I be set ;
let X, Y be ManySortedSet of I;

func X \+\ Y -> ManySortedSet of I equals :: PBOOLE:def 9
(X \ Y) \/ (Y \ X);

correctness ;
commutativity ;

end;

:: deftheorem defines \+\ PBOOLE:def 9 :

theorem Th4: :: PBOOLE:4

for I being set
for X, Y being ManySortedSet of I
for i being set st i in I holds
(X \+\ Y) . i = (X . i) \+\ (Y . i)

proof end;

theorem Th5: :: PBOOLE:5

for i, I being set holds ([[0]] I) . i = {}

proof end;

theorem Th6: :: PBOOLE:6

for I being set
for X being ManySortedSet of I st ( for i being set st i in I holds
X . i = {} ) holds
X = [[0]] I

proof end;

theorem Th7: :: PBOOLE:7

for I being set
for x, X, Y being ManySortedSet of I st ( x in X or x in Y ) holds
x in X \/ Y

proof end;

theorem Th8: :: PBOOLE:8

for I being set
for x, X, Y being ManySortedSet of I holds
( x in X /\ Y iff ( x in X & x in Y ) )

proof end;

theorem Th9: :: PBOOLE:9

for I being set
for x, X, Y being ManySortedSet of I st x in X & X c= Y holds
x in Y

proof end;

theorem Th10: :: PBOOLE:10

for I being set
for x, X, Y being ManySortedSet of I st x in X & x in Y holds
X overlaps Y

proof end;

theorem Th11: :: PBOOLE:11

for I being set
for X, Y being ManySortedSet of I st X overlaps Y holds
ex x being ManySortedSet of I st
( x in X & x in Y )

proof end;

theorem :: PBOOLE:12

for I being set
for x, X, Y being ManySortedSet of I st x in X \ Y holds
x in X

proof end;

begin

Lm1: for I being set
for X, Y being ManySortedSet of I st X c= Y & Y c= X holds
X = Y

proof end;

definition

let I be set ;
let X, Y be ManySortedSet of I;

:: original: =
redefine pred X = Y means :: PBOOLE:def 10
for i being set st i in I holds
X . i = Y . i;

compatibility by Th3;

end;

:: deftheorem defines = PBOOLE:def 10 :

theorem Th13: :: PBOOLE:13

for I being set
for X, Y, Z being ManySortedSet of I st X c= Y & Y c= Z holds
X c= Z

proof end;

theorem Th14: :: PBOOLE:14

for I being set
for X, Y being ManySortedSet of I holds X c= X \/ Y

proof end;

theorem Th15: :: PBOOLE:15

for I being set
for X, Y being ManySortedSet of I holds X /\ Y c= X

proof end;

theorem Th16: :: PBOOLE:16

for I being set
for X, Z, Y being ManySortedSet of I st X c= Z & Y c= Z holds
X \/ Y c= Z

proof end;

theorem Th17: :: PBOOLE:17

for I being set
for Z, X, Y being ManySortedSet of I st Z c= X & Z c= Y holds
Z c= X /\ Y

proof end;

theorem :: PBOOLE:18

for I being set
for X, Y, Z being ManySortedSet of I st X c= Y holds
X \/ Z c= Y \/ Z

proof end;

theorem Th19: :: PBOOLE:19

for I being set
for X, Y, Z being ManySortedSet of I st X c= Y holds
X /\ Z c= Y /\ Z

proof end;

theorem Th20: :: PBOOLE:20

for I being set
for X, Y, Z, V being ManySortedSet of I st X c= Y & Z c= V holds
X \/ Z c= Y \/ V

proof end;

theorem :: PBOOLE:21

for I being set
for X, Y, Z, V being ManySortedSet of I st X c= Y & Z c= V holds
X /\ Z c= Y /\ V

proof end;

theorem Th22: :: PBOOLE:22

for I being set
for X, Y being ManySortedSet of I st X c= Y holds
X \/ Y = Y

proof end;

theorem Th23: :: PBOOLE:23

for I being set
for X, Y being ManySortedSet of I st X c= Y holds
X /\ Y = X

proof end;

theorem :: PBOOLE:24

for I being set
for X, Y, Z being ManySortedSet of I holds X /\ Y c= X \/ Z

proof end;

theorem :: PBOOLE:25

for I being set
for X, Z, Y being ManySortedSet of I st X c= Z holds
X \/ (Y /\ Z) = (X \/ Y) /\ Z

proof end;

theorem :: PBOOLE:26

for I being set
for X, Y, Z being ManySortedSet of I holds
( X = Y \/ Z iff ( Y c= X & Z c= X & ( for V being ManySortedSet of I st Y c= V & Z c= V holds
X c= V ) ) )

proof end;

theorem :: PBOOLE:27

for I being set
for X, Y, Z being ManySortedSet of I holds
( X = Y /\ Z iff ( X c= Y & X c= Z & ( for V being ManySortedSet of I st V c= Y & V c= Z holds
V c= X ) ) )

proof end;

theorem Th28: :: PBOOLE:28

for I being set
for X, Y, Z being ManySortedSet of I holds (X \/ Y) \/ Z = X \/ (Y \/ Z)

proof end;

theorem Th29: :: PBOOLE:29

for I being set
for X, Y, Z being ManySortedSet of I holds (X /\ Y) /\ Z = X /\ (Y /\ Z)

proof end;

theorem Th30: :: PBOOLE:30

for I being set
for X, Y being ManySortedSet of I holds X /\ (X \/ Y) = X

proof end;

theorem Th31: :: PBOOLE:31

for I being set
for X, Y being ManySortedSet of I holds X \/ (X /\ Y) = X

proof end;

theorem Th32: :: PBOOLE:32

for I being set
for X, Y, Z being ManySortedSet of I holds X /\ (Y \/ Z) = (X /\ Y) \/ (X /\ Z)

proof end;

theorem Th33: :: PBOOLE:33

for I being set
for X, Y, Z being ManySortedSet of I holds X \/ (Y /\ Z) = (X \/ Y) /\ (X \/ Z)

proof end;

theorem :: PBOOLE:34

for I being set
for X, Y, Z being ManySortedSet of I st (X /\ Y) \/ (X /\ Z) = X holds
X c= Y \/ Z

proof end;

theorem :: PBOOLE:35

for I being set
for X, Y, Z being ManySortedSet of I st (X \/ Y) /\ (X \/ Z) = X holds
Y /\ Z c= X

proof end;

theorem :: PBOOLE:36

for I being set
for X, Y, Z being ManySortedSet of I holds ((X /\ Y) \/ (Y /\ Z)) \/ (Z /\ X) = ((X \/ Y) /\ (Y \/ Z)) /\ (Z \/ X)

proof end;

theorem :: PBOOLE:37

for I being set
for X, Y, Z being ManySortedSet of I st X \/ Y c= Z holds
X c= Z

proof end;

theorem :: PBOOLE:38

for I being set
for X, Y, Z being ManySortedSet of I st X c= Y /\ Z holds
X c= Y

proof end;

theorem :: PBOOLE:39

for I being set
for X, Y, Z being ManySortedSet of I holds (X \/ Y) \/ Z = (X \/ Z) \/ (Y \/ Z)

proof end;

theorem :: PBOOLE:40

for I being set
for X, Y, Z being ManySortedSet of I holds (X /\ Y) /\ Z = (X /\ Z) /\ (Y /\ Z)

proof end;

theorem :: PBOOLE:41

for I being set
for X, Y being ManySortedSet of I holds X \/ (X \/ Y) = X \/ Y

proof end;

theorem :: PBOOLE:42

for I being set
for X, Y being ManySortedSet of I holds X /\ (X /\ Y) = X /\ Y

proof end;

begin

theorem Th43: :: PBOOLE:43

for I being set
for X being ManySortedSet of I holds [[0]] I c= X

proof end;

theorem Th44: :: PBOOLE:44

for I being set
for X being ManySortedSet of I st X c= [[0]] I holds
X = [[0]] I

proof end;

theorem :: PBOOLE:45

for I being set
for X, Y, Z being ManySortedSet of I st X c= Y & X c= Z & Y /\ Z = [[0]] I holds
X = [[0]] I by Th17, Th44;

theorem :: PBOOLE:46

for I being set
for X, Y, Z being ManySortedSet of I st X c= Y & Y /\ Z = [[0]] I holds
X /\ Z = [[0]] I by Th44, Th19;

theorem :: PBOOLE:47

for I being set
for X being ManySortedSet of I holds
( X \/ ([[0]] I) = X & ([[0]] I) \/ X = X ) by Th22, Th43;

theorem :: PBOOLE:48

for I being set
for X, Y being ManySortedSet of I st X \/ Y = [[0]] I holds
X = [[0]] I by Th44, Th14;

theorem :: PBOOLE:49

for I being set
for X being ManySortedSet of I holds X /\ ([[0]] I) = [[0]] I by Th23, Th43;

theorem :: PBOOLE:50

for I being set
for X, Y, Z being ManySortedSet of I st X c= Y \/ Z & X /\ Z = [[0]] I holds
X c= Y

proof end;

theorem :: PBOOLE:51

for I being set
for Y, X being ManySortedSet of I st Y c= X & X /\ Y = [[0]] I holds
Y = [[0]] I by Th23;

begin

theorem Th52: :: PBOOLE:52

for I being set
for X, Y being ManySortedSet of I holds
( X \ Y = [[0]] I iff X c= Y )

proof end;

theorem Th53: :: PBOOLE:53

for I being set
for X, Y, Z being ManySortedSet of I st X c= Y holds
X \ Z c= Y \ Z

proof end;

theorem Th54: :: PBOOLE:54

for I being set
for X, Y, Z being ManySortedSet of I st X c= Y holds
Z \ Y c= Z \ X

proof end;

theorem :: PBOOLE:55

for I being set
for X, Y, Z, V being ManySortedSet of I st X c= Y & Z c= V holds
X \ V c= Y \ Z

proof end;

theorem Th56: :: PBOOLE:56

for I being set
for X, Y being ManySortedSet of I holds X \ Y c= X

proof end;

theorem :: PBOOLE:57

for I being set
for X, Y being ManySortedSet of I st X c= Y \ X holds
X = [[0]] I

proof end;

theorem :: PBOOLE:58

for I being set
for X being ManySortedSet of I holds X \ X = [[0]] I by Th52;

theorem Th59: :: PBOOLE:59

for I being set
for X being ManySortedSet of I holds X \ ([[0]] I) = X

proof end;

theorem Th60: :: PBOOLE:60

for I being set
for X being ManySortedSet of I holds ([[0]] I) \ X = [[0]] I

proof end;

theorem :: PBOOLE:61

for I being set
for X, Y being ManySortedSet of I holds X \ (X \/ Y) = [[0]] I

proof end;

theorem Th62: :: PBOOLE:62

for I being set
for X, Y, Z being ManySortedSet of I holds X /\ (Y \ Z) = (X /\ Y) \ Z

proof end;

theorem Th63: :: PBOOLE:63

for I being set
for X, Y being ManySortedSet of I holds (X \ Y) /\ Y = [[0]] I

proof end;

theorem Th64: :: PBOOLE:64

for I being set
for X, Y, Z being ManySortedSet of I holds X \ (Y \ Z) = (X \ Y) \/ (X /\ Z)

proof end;

theorem Th65: :: PBOOLE:65

for I being set
for X, Y being ManySortedSet of I holds (X \ Y) \/ (X /\ Y) = X

proof end;

theorem :: PBOOLE:66

for I being set
for X, Y being ManySortedSet of I st X c= Y holds
Y = X \/ (Y \ X)

proof end;

theorem Th67: :: PBOOLE:67

for I being set
for X, Y being ManySortedSet of I holds X \/ (Y \ X) = X \/ Y

proof end;

theorem Th68: :: PBOOLE:68

for I being set
for X, Y being ManySortedSet of I holds X \ (X \ Y) = X /\ Y

proof end;

theorem Th69: :: PBOOLE:69

for I being set
for X, Y, Z being ManySortedSet of I holds X \ (Y /\ Z) = (X \ Y) \/ (X \ Z)

proof end;

theorem Th70: :: PBOOLE:70

for I being set
for X, Y being ManySortedSet of I holds X \ (X /\ Y) = X \ Y

proof end;

theorem :: PBOOLE:71

for I being set
for X, Y being ManySortedSet of I holds
( X /\ Y = [[0]] I iff X \ Y = X )

proof end;

theorem Th72: :: PBOOLE:72

for I being set
for X, Y, Z being ManySortedSet of I holds (X \/ Y) \ Z = (X \ Z) \/ (Y \ Z)

proof end;

theorem Th73: :: PBOOLE:73

for I being set
for X, Y, Z being ManySortedSet of I holds (X \ Y) \ Z = X \ (Y \/ Z)

proof end;

theorem :: PBOOLE:74

for I being set
for X, Y, Z being ManySortedSet of I holds (X /\ Y) \ Z = (X \ Z) /\ (Y \ Z)

proof end;

theorem Th75: :: PBOOLE:75

for I being set
for X, Y being ManySortedSet of I holds (X \/ Y) \ Y = X \ Y

proof end;

theorem :: PBOOLE:76

for I being set
for X, Y, Z being ManySortedSet of I st X c= Y \/ Z holds
( X \ Y c= Z & X \ Z c= Y )

proof end;

theorem Th77: :: PBOOLE:77

for I being set
for X, Y being ManySortedSet of I holds (X \/ Y) \ (X /\ Y) = (X \ Y) \/ (Y \ X)

proof end;

theorem Th78: :: PBOOLE:78

for I being set
for X, Y being ManySortedSet of I holds (X \ Y) \ Y = X \ Y

proof end;

theorem :: PBOOLE:79

for I being set
for X, Y, Z being ManySortedSet of I holds X \ (Y \/ Z) = (X \ Y) /\ (X \ Z)

proof end;

theorem :: PBOOLE:80

for I being set
for X, Y being ManySortedSet of I st X \ Y = Y \ X holds
X = Y

proof end;

theorem :: PBOOLE:81

for I being set
for X, Y, Z being ManySortedSet of I holds X /\ (Y \ Z) = (X /\ Y) \ (X /\ Z)

proof end;

theorem :: PBOOLE:82

for I being set
for X, Y, Z being ManySortedSet of I st X \ Y c= Z holds
X c= Y \/ Z

proof end;

theorem :: PBOOLE:83

for I being set
for X, Y being ManySortedSet of I holds X \ Y c= X \+\ Y by Th14;

theorem :: PBOOLE:84

for I being set
for X being ManySortedSet of I holds X \+\ ([[0]] I) = X

proof end;

theorem :: PBOOLE:85

for I being set
for X being ManySortedSet of I holds X \+\ X = [[0]] I by Th52;

theorem :: PBOOLE:86

for I being set
for X, Y being ManySortedSet of I holds X \/ Y = (X \+\ Y) \/ (X /\ Y)

proof end;

theorem Th87: :: PBOOLE:87

for I being set
for X, Y being ManySortedSet of I holds X \+\ Y = (X \/ Y) \ (X /\ Y)

proof end;

theorem :: PBOOLE:88

for I being set
for X, Y, Z being ManySortedSet of I holds (X \+\ Y) \ Z = (X \ (Y \/ Z)) \/ (Y \ (X \/ Z))

proof end;

theorem :: PBOOLE:89

for I being set
for X, Y, Z being ManySortedSet of I holds X \ (Y \+\ Z) = (X \ (Y \/ Z)) \/ ((X /\ Y) /\ Z)

proof end;

theorem Th90: :: PBOOLE:90

for I being set
for X, Y, Z being ManySortedSet of I holds (X \+\ Y) \+\ Z = X \+\ (Y \+\ Z)

proof end;

theorem :: PBOOLE:91

for I being set
for X, Y, Z being ManySortedSet of I st X \ Y c= Z & Y \ X c= Z holds
X \+\ Y c= Z by Th16;

theorem Th92: :: PBOOLE:92

for I being set
for X, Y being ManySortedSet of I holds X \/ Y = X \+\ (Y \ X)

proof end;

theorem Th93: :: PBOOLE:93

for I being set
for X, Y being ManySortedSet of I holds X /\ Y = X \+\ (X \ Y)

proof end;

theorem Th94: :: PBOOLE:94

for I being set
for X, Y being ManySortedSet of I holds X \ Y = X \+\ (X /\ Y)

proof end;

theorem Th95: :: PBOOLE:95

for I being set
for Y, X being ManySortedSet of I holds Y \ X = X \+\ (X \/ Y)

proof end;

theorem :: PBOOLE:96

for I being set
for X, Y being ManySortedSet of I holds X \/ Y = (X \+\ Y) \+\ (X /\ Y)

proof end;

theorem :: PBOOLE:97

for I being set
for X, Y being ManySortedSet of I holds X /\ Y = (X \+\ Y) \+\ (X \/ Y)

proof end;

begin

theorem :: PBOOLE:98

for I being set
for X, Y, Z being ManySortedSet of I st ( X overlaps Y or X overlaps Z ) holds
X overlaps Y \/ Z

proof end;

theorem Th99: :: PBOOLE:99

for I being set
for X, Y, Z being ManySortedSet of I st X overlaps Y & Y c= Z holds
X overlaps Z

proof end;

theorem Th100: :: PBOOLE:100

for I being set
for X, Y, Z being ManySortedSet of I st X overlaps Y & X c= Z holds
Z overlaps Y

proof end;

theorem Th101: :: PBOOLE:101

for I being set
for X, Y, Z, V being ManySortedSet of I st X c= Y & Z c= V & X overlaps Z holds
Y overlaps V

proof end;

theorem :: PBOOLE:102

for I being set
for X, Y, Z being ManySortedSet of I st X overlaps Y /\ Z holds
( X overlaps Y & X overlaps Z )

proof end;

theorem :: PBOOLE:103

for I being set
for X, Z, V being ManySortedSet of I st X overlaps Z & X c= V holds
X overlaps Z /\ V

proof end;

theorem :: PBOOLE:104

for I being set
for X, Y, Z being ManySortedSet of I st X overlaps Y \ Z holds
X overlaps Y by Th56, Th99;

theorem :: PBOOLE:105

for I being set
for Y, Z, X being ManySortedSet of I st not Y overlaps Z holds
not X /\ Y overlaps X /\ Z

proof end;

theorem :: PBOOLE:106

for I being set
for X, Y, Z being ManySortedSet of I st X overlaps Y \ Z holds
Y overlaps X \ Z

proof end;

theorem Th107: :: PBOOLE:107

for I being set
for X, Y, Z being ManySortedSet of I st X meets Y & Y c= Z holds
X meets Z

proof end;

theorem Th108: :: PBOOLE:108

for I being set
for Y, X being ManySortedSet of I holds Y misses X \ Y

proof end;

theorem :: PBOOLE:109

for I being set
for X, Y being ManySortedSet of I holds X /\ Y misses X \ Y

proof end;

theorem :: PBOOLE:110

for I being set
for X, Y being ManySortedSet of I holds X /\ Y misses X \+\ Y

proof end;

theorem Th111: :: PBOOLE:111

for I being set
for X, Y being ManySortedSet of I st X misses Y holds
X /\ Y = [[0]] I

proof end;

theorem :: PBOOLE:112

for I being set
for X being ManySortedSet of I st X <> [[0]] I holds
X meets X

proof end;

theorem :: PBOOLE:113

for I being set
for X, Y, Z being ManySortedSet of I st X c= Y & X c= Z & Y misses Z holds
X = [[0]] I

proof end;

theorem :: PBOOLE:114

for I being set
for Z, V, X, Y being ManySortedSet of I st Z \/ V = X \/ Y & X misses Z & Y misses V holds
( X = V & Y = Z )

proof end;

theorem Th115: :: PBOOLE:115

for I being set
for X, Y being ManySortedSet of I st X misses Y holds
X \ Y = X

proof end;

theorem :: PBOOLE:116

for I being set
for X, Y being ManySortedSet of I st X misses Y holds
(X \/ Y) \ Y = X

proof end;

theorem :: PBOOLE:117

for I being set
for X, Y being ManySortedSet of I st X \ Y = X holds
X misses Y

proof end;

theorem :: PBOOLE:118

for I being set
for X, Y being ManySortedSet of I holds X \ Y misses Y \ X

proof end;

begin

definition

let I be set ;
let X, Y be ManySortedSet of I;

pred X [= Y means :Def11: :: PBOOLE:def 11
for x being ManySortedSet of I st x in X holds
x in Y;

reflexivity ;

end;

:: deftheorem Def11 defines [= PBOOLE:def 11 :

theorem :: PBOOLE:119

for I being set
for X, Y being ManySortedSet of I st X c= Y holds
X [= Y

proof end;

theorem :: PBOOLE:120

for I being set
for X, Y, Z being ManySortedSet of I st X [= Y & Y [= Z holds
X [= Z

proof end;

begin

theorem :: PBOOLE:121

[[0]] {} in [[0]] {}

proof end;

theorem :: PBOOLE:122

for X being ManySortedSet of {} holds X = {}

proof end;

theorem :: PBOOLE:123

for I being non empty set
for X, Y being ManySortedSet of I st X overlaps Y holds
X meets Y

proof end;

theorem Th124: :: PBOOLE:124

for I being non empty set
for x being ManySortedSet of I holds not x in [[0]] I

proof end;

theorem :: PBOOLE:125

for I being non empty set
for x, X, Y being ManySortedSet of I st x in X & x in Y holds
X /\ Y <> [[0]] I

proof end;

theorem :: PBOOLE:126

for I being non empty set
for X being ManySortedSet of I holds not X overlaps [[0]] I

proof end;

theorem Th127: :: PBOOLE:127

for I being non empty set
for X, Y being ManySortedSet of I st X /\ Y = [[0]] I holds
not X overlaps Y

proof end;

theorem :: PBOOLE:128

for I being non empty set
for X being ManySortedSet of I st X overlaps X holds
X <> [[0]] I

proof end;

begin

definition

let I be set ;
let X be ManySortedSet of I;

:: original: empty-yielding
redefine attr X is empty-yielding means :: PBOOLE:def 12
for i being set st i in I holds
X . i is empty ;

compatibility

proof end;

:: original: non-empty
redefine attr X is non-empty means :Def13: :: PBOOLE:def 13
for i being set st i in I holds
not X . i is empty ;

compatibility

proof end;

end;

:: deftheorem defines empty-yielding PBOOLE:def 12 :

:: deftheorem Def13 defines non-empty PBOOLE:def 13 :

registration

let I be set ;

cluster Relation-like V9() I -defined Function-like total for set ;

existence

proof end;

cluster Relation-like V8() I -defined Function-like total for set ;

existence

proof end;

end;

registration

let I be non empty set ;

cluster Relation-like V8() I -defined Function-like total -> V9() for set ;

coherence

proof end;

cluster Relation-like V9() I -defined Function-like total -> V8() for set ;

coherence ;

end;

theorem :: PBOOLE:129

for I being set
for X being ManySortedSet of I holds
( X is V9() iff X = [[0]] I )

proof end;

theorem :: PBOOLE:130

for I being set
for Y, X being ManySortedSet of I st Y is V9() & X c= Y holds
X is V9()

proof end;

theorem Th131: :: PBOOLE:131

for I being set
for X, Y being ManySortedSet of I st X is V8() & X c= Y holds
Y is V8()

proof end;

theorem Th132: :: PBOOLE:132

for I being set
for X, Y being ManySortedSet of I st X is V8() & X [= Y holds
X c= Y

proof end;

theorem :: PBOOLE:133

for I being set
for X, Y being ManySortedSet of I st X is V8() & X [= Y holds
Y is V8()

proof end;

theorem :: PBOOLE:134

for I being set
for X being V8() ManySortedSet of I ex x being ManySortedSet of I st x in X

proof end;

theorem Th135: :: PBOOLE:135

for I being set
for Y being ManySortedSet of I
for X being V8() ManySortedSet of I st ( for x being ManySortedSet of I holds
( x in X iff x in Y ) ) holds
X = Y

proof end;

theorem :: PBOOLE:136

for I being set
for Y, Z being ManySortedSet of I
for X being V8() ManySortedSet of I st ( for x being ManySortedSet of I holds
( x in X iff ( x in Y & x in Z ) ) ) holds
X = Y /\ Z

proof end;

begin

scheme :: PBOOLE:sch 3
MSSEx{ F₁() -> set , P₁[ set , set ] } :

ex f being ManySortedSet of F₁() st
for i being set st i in F₁() holds
P₁[i,f . i]

provided

A1: for i being set st i in F₁() holds
ex j being set st P₁[i,j]

proof end;

scheme :: PBOOLE:sch 4
MSSLambda{ F₁() -> set , F₂( set ) -> set } :

ex f being ManySortedSet of F₁() st
for i being set st i in F₁() holds
f . i = F₂(i)

proof end;

registration

let I be set ;

cluster Relation-like I -defined Function-like total Function-yielding for set ;

existence

proof end;

end;

definition

let I be set ;
mode ManySortedFunction of I is Function-yielding ManySortedSet of I;

end;

theorem :: PBOOLE:137

for I being set
for M being V8() ManySortedSet of I holds not {} in rng M

proof end;

definition

let I be set ;
let M be ManySortedSet of I;
mode Component of M is Element of rng M;

end;

theorem :: PBOOLE:138

for I being non empty set
for M being ManySortedSet of I
for A being Component of M ex i being set st
( i in I & A = M . i )

proof end;

theorem :: PBOOLE:139

for I being set
for M being ManySortedSet of I
for i being set st i in I holds
M . i is Component of M

proof end;

definition

let I be set ;
let B be ManySortedSet of I;

mode Element of B -> ManySortedSet of I means :: PBOOLE:def 14
for i being set st i in I holds
it . i is Element of B . i;

existence

proof end;

end;

:: deftheorem defines Element PBOOLE:def 14 :

begin

definition

let I be set ;
let A, B be ManySortedSet of I;

mode ManySortedFunction of A,B -> ManySortedSet of I means :Def15: :: PBOOLE:def 15
for i being set st i in I holds
it . i is Function of (A . i),(B . i);

correctness

proof end;

end;

:: deftheorem Def15 defines ManySortedFunction PBOOLE:def 15 :

registration

let I be set ;
let A, B be ManySortedSet of I;

cluster -> Function-yielding for ManySortedFunction of A,B;

coherence

proof end;

end;

registration

let I be set ;
let J be non empty set ;
let O be Function of I,J;
let F be ManySortedSet of J;

cluster O * F -> I -defined total for I -defined Function;

coherence

proof end;

end;

scheme :: PBOOLE:sch 5
LambdaDMS{ F₁() -> non empty set , F₂( set ) -> set } :

ex X being ManySortedSet of F₁() st
for d being Element of F₁() holds X . d = F₂(d)

proof end;

registration

let J be non empty set ;
let B be V8() ManySortedSet of J;
let j be Element of J;

cluster B . j -> non empty ;

coherence by Def13;

end;

definition

let I be set ;
let X, Y be ManySortedSet of I;

func [|X,Y|] -> ManySortedSet of I means :: PBOOLE:def 16
for i being set st i in I holds
it . i = [:(X . i),(Y . i):];

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem defines [| PBOOLE:def 16 :

definition

let I be set ;
let X, Y be ManySortedSet of I;
deffunc H₁( set ) -> set = Funcs ((X . $1),(Y . $1));

func (Funcs) (X,Y) -> ManySortedSet of I means :: PBOOLE:def 17
for i being set st i in I holds
it . i = Funcs ((X . i),(Y . i));

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem defines (Funcs) PBOOLE:def 17 :

definition

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

mode ManySortedSubset of M -> ManySortedSet of I means :Def18: :: PBOOLE:def 18
it c= M;

existence ;

end;

:: deftheorem Def18 defines ManySortedSubset PBOOLE:def 18 :

registration

let I be set ;
let M be V8() ManySortedSet of I;

cluster Relation-like V8() I -defined Function-like total for ManySortedSubset of M;

existence

proof end;

end;

definition

let F, G be Function-yielding Function;

func G ** F -> Function means :Def19: :: PBOOLE:def 19
( dom it = (dom F) /\ (dom G) & ( for i being set st i in dom it holds
it . i = (G . i) * (F . i) ) );