:: Categories without Uniqueness of { \bf cod } and { \bf dom }
:: by Andrzej Trybulec
::
:: Received February 28, 1995
:: Copyright (c) 1995 Association of Mizar Users

begin

registration

let A be functional set ;
cluster -> functional Element of bool A;
coherence
for b₁ being Subset of A holds b₁ is functional ;

end;

registration

let f be Function;
cluster {f} -> functional ;
coherence
{f} is functional ;

end;

theorem :: ALTCAT_1:1

canceled;

theorem Th2: :: ALTCAT_1:2

for A being set holds id A in Funcs (A,A)

proof end;

theorem Th3: :: ALTCAT_1:3

Funcs ({},{}) = {(id {})}

proof end;

theorem Th4: :: ALTCAT_1:4

for A, B, C being set
for f, g being Function st f in Funcs (A,B) & g in Funcs (B,C) holds
g * f in Funcs (A,C)

proof end;

theorem Th5: :: ALTCAT_1:5

for A, B, C being set st Funcs (A,B) <> {} & Funcs (B,C) <> {} holds
Funcs (A,C) <> {}

proof end;

theorem :: ALTCAT_1:6

canceled;

theorem :: ALTCAT_1:7

for A, B being set
for F being ManySortedSet of [:B,A:]
for C being Subset of A
for D being Subset of B
for x, y being set st x in C & y in D holds
F . (y,x) = (F | [:D,C:]) . (y,x)

proof end;

scheme :: ALTCAT_1:sch 1
MSSLambda2{ F₁() -> set , F₂() -> set , F₃( set , set ) -> set } :

ex M being ManySortedSet of [:F₁(),F₂():] st
for i, j being set st i in F₁() & j in F₂() holds
M . (i,j) = F₃(i,j)

proof end;

scheme :: ALTCAT_1:sch 2
MSSLambda2D{ F₁() -> non empty set , F₂() -> non empty set , F₃( set , set ) -> set } :

ex M being ManySortedSet of [:F₁(),F₂():] st
for i being Element of F₁()
for j being Element of F₂() holds M . (i,j) = F₃(i,j)

proof end;

scheme :: ALTCAT_1:sch 3
MSSLambda3{ F₁() -> set , F₂() -> set , F₃() -> set , F₄( set , set , set ) -> set } :

ex M being ManySortedSet of [:F₁(),F₂(),F₃():] st
for i, j, k being set st i in F₁() & j in F₂() & k in F₃() holds
M . (i,j,k) = F₄(i,j,k)

proof end;

scheme :: ALTCAT_1:sch 4
MSSLambda3D{ F₁() -> non empty set , F₂() -> non empty set , F₃() -> non empty set , F₄( set , set , set ) -> set } :

ex M being ManySortedSet of [:F₁(),F₂(),F₃():] st
for i being Element of F₁()
for j being Element of F₂()
for k being Element of F₃() holds M . (i,j,k) = F₄(i,j,k)

proof end;

theorem Th8: :: ALTCAT_1:8

for A, B being set
for N, M being ManySortedSet of [:A,B:] st ( for i, j being set st i in A & j in B holds
N . (i,j) = M . (i,j) ) holds
M = N

proof end;

theorem Th9: :: ALTCAT_1:9

for A, B being non empty set
for N, M being ManySortedSet of [:A,B:] st ( for i being Element of A
for j being Element of B holds N . (i,j) = M . (i,j) ) holds
M = N

proof end;

theorem Th10: :: ALTCAT_1:10

for A being set
for N, M being ManySortedSet of [:A,A,A:] st ( for i, j, k being set st i in A & j in A & k in A holds
N . (i,j,k) = M . (i,j,k) ) holds
M = N

proof end;

theorem :: ALTCAT_1:11

for i, j, k being set holds (i,j) :-> k = [i,j] .--> k ;

theorem Th12: :: ALTCAT_1:12

for i, j, k being set holds ((i,j) :-> k) . (i,j) = k

proof end;

begin

definition

attr c₁ is strict ;
struct AltGraph -> 1-sorted ;
aggr AltGraph(# carrier, Arrows #) -> AltGraph ;
sel Arrows c₁ -> ManySortedSet of [: the carrier of c₁, the carrier of c₁:];

end;

definition

let G be AltGraph ;
mode object of G is Element of G;

end;

definition

let G be AltGraph ;
let o1, o2 be object of G;
canceled;
func <^o1,o2^> -> set equals :: ALTCAT_1:def 2
the Arrows of G . (o1,o2);
correctness
coherence
the Arrows of G . (o1,o2) is set ;
;

end;

:: deftheorem ALTCAT_1:def 1 :
canceled;

:: deftheorem defines <^ ALTCAT_1:def 2 :
for G being AltGraph
for o1, o2 being object of G holds <^o1,o2^> = the Arrows of G . (o1,o2);

definition

let G be AltGraph ;
let o1, o2 be object of G;
mode Morphism of o1,o2 is Element of <^o1,o2^>;

end;

definition

let G be AltGraph ;
canceled;
attr G is transitive means :Def4: :: ALTCAT_1:def 4
for o1, o2, o3 being object of G st <^o1,o2^> <> {} & <^o2,o3^> <> {} holds
<^o1,o3^> <> {} ;

end;

:: deftheorem ALTCAT_1:def 3 :
canceled;

:: deftheorem Def4 defines transitive ALTCAT_1:def 4 :
for G being AltGraph holds
( G is transitive iff for o1, o2, o3 being object of G st <^o1,o2^> <> {} & <^o2,o3^> <> {} holds
<^o1,o3^> <> {} );

begin

definition

let I be set ;
let G be ManySortedSet of [:I,I:];
func {|G|} -> ManySortedSet of [:I,I,I:] means :Def5: :: ALTCAT_1:def 5
for i, j, k being set st i in I & j in I & k in I holds
it . (i,j,k) = G . (i,k);
existence
ex b₁ being ManySortedSet of [:I,I,I:] st
for i, j, k being set st i in I & j in I & k in I holds
b₁ . (i,j,k) = G . (i,k)

proof end;

uniqueness
for b₁, b₂ being ManySortedSet of [:I,I,I:] st ( for i, j, k being set st i in I & j in I & k in I holds
b₁ . (i,j,k) = G . (i,k) ) & ( for i, j, k being set st i in I & j in I & k in I holds
b₂ . (i,j,k) = G . (i,k) ) holds
b₁ = b₂

proof end;

let H be ManySortedSet of [:I,I:];
func {|G,H|} -> ManySortedSet of [:I,I,I:] means :Def6: :: ALTCAT_1:def 6
for i, j, k being set st i in I & j in I & k in I holds
it . (i,j,k) = [:(H . (j,k)),(G . (i,j)):];
existence
ex b₁ being ManySortedSet of [:I,I,I:] st
for i, j, k being set st i in I & j in I & k in I holds
b₁ . (i,j,k) = [:(H . (j,k)),(G . (i,j)):]

proof end;

uniqueness
for b₁, b₂ being ManySortedSet of [:I,I,I:] st ( for i, j, k being set st i in I & j in I & k in I holds
b₁ . (i,j,k) = [:(H . (j,k)),(G . (i,j)):] ) & ( for i, j, k being set st i in I & j in I & k in I holds
b₂ . (i,j,k) = [:(H . (j,k)),(G . (i,j)):] ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def5 defines {| ALTCAT_1:def 5 :
for I being set
for G being ManySortedSet of [:I,I:]
for b₃ being ManySortedSet of [:I,I,I:] holds
( b₃ = {|G|} iff for i, j, k being set st i in I & j in I & k in I holds
b₃ . (i,j,k) = G . (i,k) );

:: deftheorem Def6 defines {| ALTCAT_1:def 6 :
for I being set
for G, H being ManySortedSet of [:I,I:]
for b₄ being ManySortedSet of [:I,I,I:] holds
( b₄ = {|G,H|} iff for i, j, k being set st i in I & j in I & k in I holds
b₄ . (i,j,k) = [:(H . (j,k)),(G . (i,j)):] );

definition

let I be set ;
let G be ManySortedSet of [:I,I:];
mode BinComp of G is ManySortedFunction of {|G,G|},{|G|};

end;

definition

let I be non empty set ;
let G be ManySortedSet of [:I,I:];
let o be BinComp of G;
let i, j, k be Element of I;
:: original: .
redefine func o . (i,j,k) -> Function of [:(G . (j,k)),(G . (i,j)):],(G . (i,k));
coherence
o . (i,j,k) is Function of [:(G . (j,k)),(G . (i,j)):],(G . (i,k))

proof end;

end;

definition

let I be non empty set ;
let G be ManySortedSet of [:I,I:];
let IT be BinComp of G;
attr IT is associative means :Def7: :: ALTCAT_1:def 7
for i, j, k, l being Element of I
for f, g, h being set st f in G . (i,j) & g in G . (j,k) & h in G . (k,l) holds
(IT . (i,k,l)) . (h,((IT . (i,j,k)) . (g,f))) = (IT . (i,j,l)) . (((IT . (j,k,l)) . (h,g)),f);
attr IT is with_right_units means :Def8: :: ALTCAT_1:def 8
for i being Element of I ex e being set st
( e in G . (i,i) & ( for j being Element of I
for f being set st f in G . (i,j) holds
(IT . (i,i,j)) . (f,e) = f ) );
attr IT is with_left_units means :Def9: :: ALTCAT_1:def 9
for j being Element of I ex e being set st
( e in G . (j,j) & ( for i being Element of I
for f being set st f in G . (i,j) holds
(IT . (i,j,j)) . (e,f) = f ) );

end;

:: deftheorem Def7 defines associative ALTCAT_1:def 7 :
for I being non empty set
for G being ManySortedSet of [:I,I:]
for IT being BinComp of G holds
( IT is associative iff for i, j, k, l being Element of I
for f, g, h being set st f in G . (i,j) & g in G . (j,k) & h in G . (k,l) holds
(IT . (i,k,l)) . (h,((IT . (i,j,k)) . (g,f))) = (IT . (i,j,l)) . (((IT . (j,k,l)) . (h,g)),f) );

:: deftheorem Def8 defines with_right_units ALTCAT_1:def 8 :
for I being non empty set
for G being ManySortedSet of [:I,I:]
for IT being BinComp of G holds
( IT is with_right_units iff for i being Element of I ex e being set st
( e in G . (i,i) & ( for j being Element of I
for f being set st f in G . (i,j) holds
(IT . (i,i,j)) . (f,e) = f ) ) );

:: deftheorem Def9 defines with_left_units ALTCAT_1:def 9 :
for I being non empty set
for G being ManySortedSet of [:I,I:]
for IT being BinComp of G holds
( IT is with_left_units iff for j being Element of I ex e being set st
( e in G . (j,j) & ( for i being Element of I
for f being set st f in G . (i,j) holds
(IT . (i,j,j)) . (e,f) = f ) ) );

begin

definition

attr c₁ is strict ;
struct AltCatStr -> AltGraph ;
aggr AltCatStr(# carrier, Arrows, Comp #) -> AltCatStr ;
sel Comp c₁ -> BinComp of the Arrows of c₁;

end;

registration

cluster non empty strict AltCatStr ;
existence
ex b₁ being AltCatStr st
( b₁ is strict & not b₁ is empty )

proof end;

end;

definition

let C be non empty AltCatStr ;
let o1, o2, o3 be object of C;
assume A1: ( <^o1,o2^> <> {} & <^o2,o3^> <> {} ) ;
let f be Morphism of o1,o2;
let g be Morphism of o2,o3;
func g * f -> Morphism of o1,o3 equals :Def10: :: ALTCAT_1:def 10
( the Comp of C . (o1,o2,o3)) . (g,f);
coherence
( the Comp of C . (o1,o2,o3)) . (g,f) is Morphism of o1,o3

proof end;

correctness
;

end;

:: deftheorem Def10 defines * ALTCAT_1:def 10 :
for C being non empty AltCatStr
for o1, o2, o3 being object of C st <^o1,o2^> <> {} & <^o2,o3^> <> {} holds
for f being Morphism of o1,o2
for g being Morphism of o2,o3 holds g * f = ( the Comp of C . (o1,o2,o3)) . (g,f);

definition

let IT be Function;
attr IT is compositional means :Def11: :: ALTCAT_1:def 11
for x being set st x in dom IT holds
ex f, g being Function st
( x = [g,f] & IT . x = g * f );

end;

:: deftheorem Def11 defines compositional ALTCAT_1:def 11 :
for IT being Function holds
( IT is compositional iff for x being set st x in dom IT holds
ex f, g being Function st
( x = [g,f] & IT . x = g * f ) );

registration

let A, B be functional set ;
cluster Relation-like [:A,B:] -defined Function-like V14([:A,B:]) Function-yielding compositional set ;
existence
ex b₁ being ManySortedFunction of [:A,B:] st b₁ is compositional

proof end;

end;

theorem Th13: :: ALTCAT_1:13

for A, B being functional set
for F being compositional ManySortedSet of [:A,B:]
for g, f being Function st g in A & f in B holds
F . (g,f) = g * f

proof end;

definition

let A, B be functional set ;
func FuncComp (A,B) -> compositional ManySortedFunction of [:B,A:] means :Def12: :: ALTCAT_1:def 12
verum;
uniqueness
for b₁, b₂ being compositional ManySortedFunction of [:B,A:] holds b₁ = b₂

proof end;

correctness
existence
ex b₁ being compositional ManySortedFunction of [:B,A:] st verum;
;

end;

:: deftheorem Def12 defines FuncComp ALTCAT_1:def 12 :
for A, B being functional set
for b₃ being compositional ManySortedFunction of [:B,A:] holds
( b₃ = FuncComp (A,B) iff verum );

theorem Th14: :: ALTCAT_1:14

for A, B, C being set holds rng (FuncComp ((Funcs (A,B)),(Funcs (B,C)))) c= Funcs (A,C)

proof end;

theorem Th15: :: ALTCAT_1:15

for o being set holds FuncComp ({(id o)},{(id o)}) = ((id o),(id o)) :-> (id o)

proof end;

theorem Th16: :: ALTCAT_1:16

for A, B being functional set
for A1 being Subset of A
for B1 being Subset of B holds FuncComp (A1,B1) = (FuncComp (A,B)) | [:B1,A1:]

proof end;

definition

let C be non empty AltCatStr ;
attr C is quasi-functional means :Def13: :: ALTCAT_1:def 13
for a1, a2 being object of C holds <^a1,a2^> c= Funcs (a1,a2);
attr C is semi-functional means :Def14: :: ALTCAT_1:def 14
for a1, a2, a3 being object of C st <^a1,a2^> <> {} & <^a2,a3^> <> {} & <^a1,a3^> <> {} holds
for f being Morphism of a1,a2
for g being Morphism of a2,a3
for f9, g9 being Function st f = f9 & g = g9 holds
g * f = g9 * f9;
attr C is pseudo-functional means :Def15: :: ALTCAT_1:def 15
for o1, o2, o3 being object of C holds the Comp of C . (o1,o2,o3) = (FuncComp ((Funcs (o1,o2)),(Funcs (o2,o3)))) | [:<^o2,o3^>,<^o1,o2^>:];

end;

:: deftheorem Def13 defines quasi-functional ALTCAT_1:def 13 :
for C being non empty AltCatStr holds
( C is quasi-functional iff for a1, a2 being object of C holds <^a1,a2^> c= Funcs (a1,a2) );

:: deftheorem Def14 defines semi-functional ALTCAT_1:def 14 :
for C being non empty AltCatStr holds
( C is semi-functional iff for a1, a2, a3 being object of C st <^a1,a2^> <> {} & <^a2,a3^> <> {} & <^a1,a3^> <> {} holds
for f being Morphism of a1,a2
for g being Morphism of a2,a3
for f9, g9 being Function st f = f9 & g = g9 holds
g * f = g9 * f9 );

:: deftheorem Def15 defines pseudo-functional ALTCAT_1:def 15 :
for C being non empty AltCatStr holds
( C is pseudo-functional iff for o1, o2, o3 being object of C holds the Comp of C . (o1,o2,o3) = (FuncComp ((Funcs (o1,o2)),(Funcs (o2,o3)))) | [:<^o2,o3^>,<^o1,o2^>:] );

registration

let X be non empty set ;
let A be ManySortedSet of [:X,X:];
let C be BinComp of A;
cluster AltCatStr(# X,A,C #) -> non empty ;
coherence
not AltCatStr(# X,A,C #) is empty ;

end;

registration

cluster non empty strict pseudo-functional AltCatStr ;
existence
ex b₁ being non empty AltCatStr st
( b₁ is strict & b₁ is pseudo-functional )

proof end;

end;

theorem :: ALTCAT_1:17

for C being non empty AltCatStr
for a1, a2, a3 being object of C holds the Comp of C . (a1,a2,a3) is Function of [:<^a2,a3^>,<^a1,a2^>:],<^a1,a3^> ;

theorem Th18: :: ALTCAT_1:18

for C being non empty pseudo-functional AltCatStr
for a1, a2, a3 being object of C st <^a1,a2^> <> {} & <^a2,a3^> <> {} & <^a1,a3^> <> {} holds
for f being Morphism of a1,a2
for g being Morphism of a2,a3
for f9, g9 being Function st f = f9 & g = g9 holds
g * f = g9 * f9

proof end;

definition

let A be non empty set ;
func EnsCat A -> non empty strict pseudo-functional AltCatStr means :Def16: :: ALTCAT_1:def 16
( the carrier of it = A & ( for a1, a2 being object of it holds <^a1,a2^> = Funcs (a1,a2) ) );
existence
ex b₁ being non empty strict pseudo-functional AltCatStr st
( the carrier of b₁ = A & ( for a1, a2 being object of b₁ holds <^a1,a2^> = Funcs (a1,a2) ) )

proof end;

uniqueness
for b₁, b₂ being non empty strict pseudo-functional AltCatStr st the carrier of b₁ = A & ( for a1, a2 being object of b₁ holds <^a1,a2^> = Funcs (a1,a2) ) & the carrier of b₂ = A & ( for a1, a2 being object of b₂ holds <^a1,a2^> = Funcs (a1,a2) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def16 defines EnsCat ALTCAT_1:def 16 :
for A being non empty set
for b₂ being non empty strict pseudo-functional AltCatStr holds
( b₂ = EnsCat A iff ( the carrier of b₂ = A & ( for a1, a2 being object of b₂ holds <^a1,a2^> = Funcs (a1,a2) ) ) );

definition

let C be non empty AltCatStr ;
attr C is associative means :Def17: :: ALTCAT_1:def 17
the Comp of C is associative ;
attr C is with_units means :Def18: :: ALTCAT_1:def 18
( the Comp of C is with_left_units & the Comp of C is with_right_units );

end;

:: deftheorem Def17 defines associative ALTCAT_1:def 17 :
for C being non empty AltCatStr holds
( C is associative iff the Comp of C is associative );

:: deftheorem Def18 defines with_units ALTCAT_1:def 18 :
for C being non empty AltCatStr holds
( C is with_units iff ( the Comp of C is with_left_units & the Comp of C is with_right_units ) );

Lm1: for A being non empty set holds
( EnsCat A is transitive & EnsCat A is associative & EnsCat A is with_units )

proof end;

registration

cluster non empty transitive strict associative with_units AltCatStr ;
existence
ex b₁ being non empty AltCatStr st
( b₁ is transitive & b₁ is associative & b₁ is with_units & b₁ is strict )

proof end;

end;

theorem :: ALTCAT_1:19

canceled;

theorem :: ALTCAT_1:20

for C being non empty transitive AltCatStr
for a1, a2, a3 being object of C holds
( dom ( the Comp of C . (a1,a2,a3)) = [:<^a2,a3^>,<^a1,a2^>:] & rng ( the Comp of C . (a1,a2,a3)) c= <^a1,a3^> )

proof end;

theorem Th21: :: ALTCAT_1:21

for C being non empty with_units AltCatStr
for o being object of C holds <^o,o^> <> {}

proof end;

registration

let A be non empty set ;
cluster EnsCat A -> non empty transitive strict pseudo-functional associative with_units ;
coherence
( EnsCat A is transitive & EnsCat A is associative & EnsCat A is with_units ) by Lm1;

end;

registration

cluster non empty transitive quasi-functional semi-functional -> non empty pseudo-functional AltCatStr ;
coherence
for b₁ being non empty AltCatStr st b₁ is quasi-functional & b₁ is semi-functional & b₁ is transitive holds
b₁ is pseudo-functional

proof end;

cluster non empty transitive pseudo-functional with_units -> non empty quasi-functional semi-functional AltCatStr ;
coherence
for b₁ being non empty AltCatStr st b₁ is with_units & b₁ is pseudo-functional & b₁ is transitive holds
( b₁ is quasi-functional & b₁ is semi-functional )

proof end;

end;

definition

mode category is non empty transitive associative with_units AltCatStr ;

end;

begin

definition

let C be non empty with_units AltCatStr ;
let o be object of C;
func idm o -> Morphism of o,o means :Def19: :: ALTCAT_1:def 19
for o9 being object of C st <^o,o9^> <> {} holds
for a being Morphism of o,o9 holds a * it = a;
existence
ex b₁ being Morphism of o,o st
for o9 being object of C st <^o,o9^> <> {} holds
for a being Morphism of o,o9 holds a * b₁ = a

proof end;

uniqueness
for b₁, b₂ being Morphism of o,o st ( for o9 being object of C st <^o,o9^> <> {} holds
for a being Morphism of o,o9 holds a * b₁ = a ) & ( for o9 being object of C st <^o,o9^> <> {} holds
for a being Morphism of o,o9 holds a * b₂ = a ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def19 defines idm ALTCAT_1:def 19 :
for C being non empty with_units AltCatStr
for o being object of C
for b₃ being Morphism of o,o holds
( b₃ = idm o iff for o9 being object of C st <^o,o9^> <> {} holds
for a being Morphism of o,o9 holds a * b₃ = a );

theorem :: ALTCAT_1:22

canceled;

theorem Th23: :: ALTCAT_1:23

for C being non empty with_units AltCatStr
for o being object of C holds idm o in <^o,o^>

proof end;

theorem :: ALTCAT_1:24

for C being non empty with_units AltCatStr
for o1, o2 being object of C st <^o1,o2^> <> {} holds
for a being Morphism of o1,o2 holds (idm o2) * a = a

proof end;

theorem :: ALTCAT_1:25

for C being non empty transitive associative AltCatStr
for o1, o2, o3, o4 being object of C st <^o1,o2^> <> {} & <^o2,o3^> <> {} & <^o3,o4^> <> {} holds
for a being Morphism of o1,o2
for b being Morphism of o2,o3
for c being Morphism of o3,o4 holds c * (b * a) = (c * b) * a

proof end;

begin

definition

let C be AltCatStr ;
attr C is quasi-discrete means :Def20: :: ALTCAT_1:def 20
for i, j being object of C st <^i,j^> <> {} holds
i = j;
attr C is pseudo-discrete means :Def21: :: ALTCAT_1:def 21
for i being object of C holds <^i,i^> is trivial ;

end;

:: deftheorem Def20 defines quasi-discrete ALTCAT_1:def 20 :
for C being AltCatStr holds
( C is quasi-discrete iff for i, j being object of C st <^i,j^> <> {} holds
i = j );

:: deftheorem Def21 defines pseudo-discrete ALTCAT_1:def 21 :
for C being AltCatStr holds
( C is pseudo-discrete iff for i being object of C holds <^i,i^> is trivial );

theorem :: ALTCAT_1:26

for C being non empty with_units AltCatStr holds
( C is pseudo-discrete iff for o being object of C holds <^o,o^> = {(idm o)} )

proof end;

registration

cluster non empty trivial -> quasi-discrete AltCatStr ;
coherence
for b₁ being AltCatStr st b₁ is trivial & not b₁ is empty holds
b₁ is quasi-discrete

proof end;

end;

theorem Th27: :: ALTCAT_1:27

( EnsCat 1 is pseudo-discrete & EnsCat 1 is trivial )

proof end;

registration

cluster non empty trivial transitive strict associative with_units pseudo-discrete AltCatStr ;
existence
ex b₁ being category st
( b₁ is pseudo-discrete & b₁ is trivial & b₁ is strict ) by Th27;

end;

registration

cluster non empty trivial transitive strict associative with_units quasi-discrete pseudo-discrete AltCatStr ;
existence
ex b₁ being category st
( b₁ is quasi-discrete & b₁ is pseudo-discrete & b₁ is trivial & b₁ is strict ) by Th27;

end;

definition

mode discrete_category is quasi-discrete pseudo-discrete category;

end;

definition

let A be non empty set ;
func DiscrCat A -> non empty strict quasi-discrete AltCatStr means :Def22: :: ALTCAT_1:def 22
( the carrier of it = A & ( for i being object of it holds <^i,i^> = {(id i)} ) );
existence
ex b₁ being non empty strict quasi-discrete AltCatStr st
( the carrier of b₁ = A & ( for i being object of b₁ holds <^i,i^> = {(id i)} ) )

proof end;

correctness
uniqueness
for b₁, b₂ being non empty strict quasi-discrete AltCatStr st the carrier of b₁ = A & ( for i being object of b₁ holds <^i,i^> = {(id i)} ) & the carrier of b₂ = A & ( for i being object of b₂ holds <^i,i^> = {(id i)} ) holds
b₁ = b₂;

proof end;

end;

:: deftheorem Def22 defines DiscrCat ALTCAT_1:def 22 :
for A being non empty set
for b₂ being non empty strict quasi-discrete AltCatStr holds
( b₂ = DiscrCat A iff ( the carrier of b₂ = A & ( for i being object of b₂ holds <^i,i^> = {(id i)} ) ) );

registration

cluster quasi-discrete -> transitive AltCatStr ;
coherence
for b₁ being AltCatStr st b₁ is quasi-discrete holds
b₁ is transitive

proof end;

end;

theorem Th28: :: ALTCAT_1:28

for A being non empty set
for o1, o2, o3 being object of (DiscrCat A) st ( o1 <> o2 or o2 <> o3 ) holds
the Comp of (DiscrCat A) . (o1,o2,o3) = {}

proof end;

theorem Th29: :: ALTCAT_1:29

for A being non empty set
for o being object of (DiscrCat A) holds the Comp of (DiscrCat A) . (o,o,o) = ((id o),(id o)) :-> (id o)

proof end;

registration

let A be non empty set ;
cluster DiscrCat A -> non empty strict pseudo-functional associative with_units quasi-discrete pseudo-discrete ;
coherence
( DiscrCat A is pseudo-functional & DiscrCat A is pseudo-discrete & DiscrCat A is with_units & DiscrCat A is associative )

proof end;

end;