begin
begin
definition
let L be ( ( ) ( )
1-sorted ) ;
let R be ( ( ) (
Relation-like the
carrier of
L : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-defined the
carrier of
L : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-valued )
Relation of ) ;
let C be ( ( ) ( )
strict_chain of
R : ( ( ) (
Relation-like the
carrier of
L : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-defined the
carrier of
L : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-valued )
Relation of ) ) ;
attr C is
maximal means
for
D being ( ( ) ( )
strict_chain of
R : ( ( ) (
Relation-like the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-valued )
Element of
bool [: the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non
empty )
set ) : ( ( ) ( non
empty )
set ) ) ) st
C : ( ( ) (
Relation-like R : ( ( ) (
Relation-like the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-valued )
Element of
bool [: the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non
empty )
set ) : ( ( ) ( non
empty )
set ) )
-defined R : ( ( ) (
Relation-like the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-valued )
Element of
bool [: the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non
empty )
set ) : ( ( ) ( non
empty )
set ) )
-valued )
Element of
bool [:R : ( ( ) ( Relation-like the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) -defined the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) -valued ) Element of bool [: the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,R : ( ( ) ( Relation-like the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) -defined the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) -valued ) Element of bool [: the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( )
set ) : ( ( ) ( non
empty )
set ) )
c= D : ( ( ) ( )
strict_chain of
R : ( ( ) (
Relation-like the
carrier of
L : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-defined the
carrier of
L : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-valued )
Relation of ) ) holds
C : ( ( ) (
Relation-like R : ( ( ) (
Relation-like the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-valued )
Element of
bool [: the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non
empty )
set ) : ( ( ) ( non
empty )
set ) )
-defined R : ( ( ) (
Relation-like the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-valued )
Element of
bool [: the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non
empty )
set ) : ( ( ) ( non
empty )
set ) )
-valued )
Element of
bool [:R : ( ( ) ( Relation-like the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) -defined the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) -valued ) Element of bool [: the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,R : ( ( ) ( Relation-like the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) -defined the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) -valued ) Element of bool [: the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( )
set ) : ( ( ) ( non
empty )
set ) )
= D : ( ( ) ( )
strict_chain of
R : ( ( ) (
Relation-like the
carrier of
L : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-defined the
carrier of
L : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-valued )
Relation of ) ) ;
end;
definition
let L be ( ( ) ( )
1-sorted ) ;
let R be ( ( ) (
Relation-like the
carrier of
L : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-defined the
carrier of
L : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-valued )
Relation of ) ;
let C be ( ( ) ( )
set ) ;
func Strict_Chains (
R,
C)
-> ( ( ) ( )
set )
means
for
x being ( ( ) ( )
set ) holds
(
x : ( ( ) ( )
set )
in it : ( (
Function-like V31(
R : ( ( ) (
Relation-like the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-valued )
Element of
bool [: the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non
empty )
set ) : ( ( ) ( non
empty )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like R : ( ( ) (
Relation-like the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-valued )
Element of
bool [: the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non
empty )
set ) : ( ( ) ( non
empty )
set ) )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-valued Function-like V31(
R : ( ( ) (
Relation-like the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-valued )
Element of
bool [: the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non
empty )
set ) : ( ( ) ( non
empty )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set ) ) )
Element of
bool [:R : ( ( ) ( Relation-like the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) -defined the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) -valued ) Element of bool [: the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( )
set ) : ( ( ) ( non
empty )
set ) ) iff (
x : ( ( ) ( )
set ) is ( ( ) ( )
strict_chain of
R : ( ( ) (
Relation-like the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-valued )
Element of
bool [: the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non
empty )
set ) : ( ( ) ( non
empty )
set ) ) ) &
C : ( ( ) (
Relation-like R : ( ( ) (
Relation-like the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-valued )
Element of
bool [: the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non
empty )
set ) : ( ( ) ( non
empty )
set ) )
-defined R : ( ( ) (
Relation-like the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-valued )
Element of
bool [: the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non
empty )
set ) : ( ( ) ( non
empty )
set ) )
-valued )
Element of
bool [:R : ( ( ) ( Relation-like the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) -defined the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) -valued ) Element of bool [: the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,R : ( ( ) ( Relation-like the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) -defined the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) -valued ) Element of bool [: the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( )
set ) : ( ( ) ( non
empty )
set ) )
c= x : ( ( ) ( )
set ) ) );
end;
theorem
for
L being ( ( ) ( )
1-sorted )
for
R being ( ( ) (
Relation-like the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-defined the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-valued )
Relation of )
for
C being ( ( ) ( )
strict_chain of
R : ( ( ) (
Relation-like the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-defined the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-valued )
Relation of ) ) holds
(
Strict_Chains (
R : ( ( ) (
Relation-like the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-defined the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-valued )
Relation of ) ,
C : ( ( ) ( )
strict_chain of
b2 : ( ( ) (
Relation-like the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-defined the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-valued )
Relation of ) ) ) : ( ( ) ( non
empty )
set )
is_inductive_wrt RelIncl (Strict_Chains (R : ( ( ) ( Relation-like the carrier of b1 : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -defined the carrier of b1 : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -valued ) Relation of ) ,C : ( ( ) ( ) strict_chain of b2 : ( ( ) ( Relation-like the carrier of b1 : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -defined the carrier of b1 : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -valued ) Relation of ) ) )) : ( ( ) ( non
empty )
set ) : ( (
V27(
Strict_Chains (
b2 : ( ( ) (
Relation-like the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-defined the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-valued )
Relation of ) ,
b3 : ( ( ) ( )
strict_chain of
b2 : ( ( ) (
Relation-like the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-defined the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-valued )
Relation of ) ) ) : ( ( ) ( non
empty )
set ) )
reflexive antisymmetric transitive ) (
Relation-like Strict_Chains (
b2 : ( ( ) (
Relation-like the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-defined the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-valued )
Relation of ) ,
b3 : ( ( ) ( )
strict_chain of
b2 : ( ( ) (
Relation-like the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-defined the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-valued )
Relation of ) ) ) : ( ( ) ( non
empty )
set )
-defined Strict_Chains (
b2 : ( ( ) (
Relation-like the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-defined the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-valued )
Relation of ) ,
b3 : ( ( ) ( )
strict_chain of
b2 : ( ( ) (
Relation-like the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-defined the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-valued )
Relation of ) ) ) : ( ( ) ( non
empty )
set )
-valued V27(
Strict_Chains (
b2 : ( ( ) (
Relation-like the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-defined the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-valued )
Relation of ) ,
b3 : ( ( ) ( )
strict_chain of
b2 : ( ( ) (
Relation-like the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-defined the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-valued )
Relation of ) ) ) : ( ( ) ( non
empty )
set ) )
reflexive antisymmetric transitive )
Element of
bool [:(Strict_Chains (b2 : ( ( ) ( Relation-like the carrier of b1 : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -defined the carrier of b1 : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -valued ) Relation of ) ,b3 : ( ( ) ( ) strict_chain of b2 : ( ( ) ( Relation-like the carrier of b1 : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -defined the carrier of b1 : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -valued ) Relation of ) ) )) : ( ( ) ( non empty ) set ) ,(Strict_Chains (b2 : ( ( ) ( Relation-like the carrier of b1 : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -defined the carrier of b1 : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -valued ) Relation of ) ,b3 : ( ( ) ( ) strict_chain of b2 : ( ( ) ( Relation-like the carrier of b1 : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -defined the carrier of b1 : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -valued ) Relation of ) ) )) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non
empty )
set ) : ( ( ) ( non
empty )
set ) ) & ex
D being ( ( ) ( )
set ) st
(
D : ( ( ) ( )
set )
is_maximal_in RelIncl (Strict_Chains (R : ( ( ) ( Relation-like the carrier of b1 : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -defined the carrier of b1 : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -valued ) Relation of ) ,C : ( ( ) ( ) strict_chain of b2 : ( ( ) ( Relation-like the carrier of b1 : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -defined the carrier of b1 : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -valued ) Relation of ) ) )) : ( ( ) ( non
empty )
set ) : ( (
V27(
Strict_Chains (
b2 : ( ( ) (
Relation-like the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-defined the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-valued )
Relation of ) ,
b3 : ( ( ) ( )
strict_chain of
b2 : ( ( ) (
Relation-like the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-defined the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-valued )
Relation of ) ) ) : ( ( ) ( non
empty )
set ) )
reflexive antisymmetric transitive ) (
Relation-like Strict_Chains (
b2 : ( ( ) (
Relation-like the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-defined the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-valued )
Relation of ) ,
b3 : ( ( ) ( )
strict_chain of
b2 : ( ( ) (
Relation-like the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-defined the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-valued )
Relation of ) ) ) : ( ( ) ( non
empty )
set )
-defined Strict_Chains (
b2 : ( ( ) (
Relation-like the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-defined the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-valued )
Relation of ) ,
b3 : ( ( ) ( )
strict_chain of
b2 : ( ( ) (
Relation-like the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-defined the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-valued )
Relation of ) ) ) : ( ( ) ( non
empty )
set )
-valued V27(
Strict_Chains (
b2 : ( ( ) (
Relation-like the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-defined the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-valued )
Relation of ) ,
b3 : ( ( ) ( )
strict_chain of
b2 : ( ( ) (
Relation-like the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-defined the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-valued )
Relation of ) ) ) : ( ( ) ( non
empty )
set ) )
reflexive antisymmetric transitive )
Element of
bool [:(Strict_Chains (b2 : ( ( ) ( Relation-like the carrier of b1 : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -defined the carrier of b1 : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -valued ) Relation of ) ,b3 : ( ( ) ( ) strict_chain of b2 : ( ( ) ( Relation-like the carrier of b1 : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -defined the carrier of b1 : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -valued ) Relation of ) ) )) : ( ( ) ( non empty ) set ) ,(Strict_Chains (b2 : ( ( ) ( Relation-like the carrier of b1 : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -defined the carrier of b1 : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -valued ) Relation of ) ,b3 : ( ( ) ( ) strict_chain of b2 : ( ( ) ( Relation-like the carrier of b1 : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -defined the carrier of b1 : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -valued ) Relation of ) ) )) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non
empty )
set ) : ( ( ) ( non
empty )
set ) ) &
C : ( ( ) ( )
strict_chain of
b2 : ( ( ) (
Relation-like the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-defined the
carrier of
b1 : ( ( ) ( )
1-sorted ) : ( ( ) ( )
set )
-valued )
Relation of ) )
c= D : ( ( ) ( )
set ) ) ) ;
theorem
for
L being ( ( non
empty reflexive transitive antisymmetric ) ( non
empty reflexive transitive antisymmetric )
Poset)
for
R being ( (
auxiliary(i) auxiliary(ii) ) (
Relation-like the
carrier of
b1 : ( ( non
empty reflexive transitive antisymmetric ) ( non
empty reflexive transitive antisymmetric )
Poset) : ( ( ) ( non
empty )
set )
-defined the
carrier of
b1 : ( ( non
empty reflexive transitive antisymmetric ) ( non
empty reflexive transitive antisymmetric )
Poset) : ( ( ) ( non
empty )
set )
-valued auxiliary(i) auxiliary(ii) )
Relation of )
for
C being ( ( non
empty ) ( non
empty )
strict_chain of
R : ( (
auxiliary(i) auxiliary(ii) ) (
Relation-like the
carrier of
b1 : ( ( non
empty reflexive transitive antisymmetric ) ( non
empty reflexive transitive antisymmetric )
Poset) : ( ( ) ( non
empty )
set )
-defined the
carrier of
b1 : ( ( non
empty reflexive transitive antisymmetric ) ( non
empty reflexive transitive antisymmetric )
Poset) : ( ( ) ( non
empty )
set )
-valued auxiliary(i) auxiliary(ii) )
Relation of ) )
for
X being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) st
ex_inf_of (uparrow ("\/" (X : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ,L : ( ( non empty reflexive transitive antisymmetric ) ( non empty reflexive transitive antisymmetric ) Poset) )) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty reflexive transitive antisymmetric ) ( non empty reflexive transitive antisymmetric ) Poset) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non
empty filtered upper )
Element of
bool the
carrier of
b1 : ( ( non
empty reflexive transitive antisymmetric ) ( non
empty reflexive transitive antisymmetric )
Poset) : ( ( ) ( non
empty )
set ) : ( ( ) ( non
empty )
set ) )
/\ C : ( ( non
empty ) ( non
empty )
strict_chain of
b2 : ( (
auxiliary(i) auxiliary(ii) ) (
Relation-like the
carrier of
b1 : ( ( non
empty reflexive transitive antisymmetric ) ( non
empty reflexive transitive antisymmetric )
Poset) : ( ( ) ( non
empty )
set )
-defined the
carrier of
b1 : ( ( non
empty reflexive transitive antisymmetric ) ( non
empty reflexive transitive antisymmetric )
Poset) : ( ( ) ( non
empty )
set )
-valued auxiliary(i) auxiliary(ii) )
Relation of ) ) : ( ( ) ( )
Element of
bool the
carrier of
b1 : ( ( non
empty reflexive transitive antisymmetric ) ( non
empty reflexive transitive antisymmetric )
Poset) : ( ( ) ( non
empty )
set ) : ( ( ) ( non
empty )
set ) ) ,
L : ( ( non
empty reflexive transitive antisymmetric ) ( non
empty reflexive transitive antisymmetric )
Poset) &
ex_sup_of X : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ,
L : ( ( non
empty reflexive transitive antisymmetric ) ( non
empty reflexive transitive antisymmetric )
Poset) &
C : ( ( non
empty ) ( non
empty )
strict_chain of
b2 : ( (
auxiliary(i) auxiliary(ii) ) (
Relation-like the
carrier of
b1 : ( ( non
empty reflexive transitive antisymmetric ) ( non
empty reflexive transitive antisymmetric )
Poset) : ( ( ) ( non
empty )
set )
-defined the
carrier of
b1 : ( ( non
empty reflexive transitive antisymmetric ) ( non
empty reflexive transitive antisymmetric )
Poset) : ( ( ) ( non
empty )
set )
-valued auxiliary(i) auxiliary(ii) )
Relation of ) ) is
maximal holds
(
"\/" (
X : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ,
(subrelstr C : ( ( non empty ) ( non empty ) strict_chain of b2 : ( ( auxiliary(i) auxiliary(ii) ) ( Relation-like the carrier of b1 : ( ( non empty reflexive transitive antisymmetric ) ( non empty reflexive transitive antisymmetric ) Poset) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty reflexive transitive antisymmetric ) ( non empty reflexive transitive antisymmetric ) Poset) : ( ( ) ( non empty ) set ) -valued auxiliary(i) auxiliary(ii) ) Relation of ) ) ) : ( (
strict V134(
b1 : ( ( non
empty reflexive transitive antisymmetric ) ( non
empty reflexive transitive antisymmetric )
Poset) ) ) ( non
empty strict reflexive transitive antisymmetric V134(
b1 : ( ( non
empty reflexive transitive antisymmetric ) ( non
empty reflexive transitive antisymmetric )
Poset) ) )
SubRelStr of
b1 : ( ( non
empty reflexive transitive antisymmetric ) ( non
empty reflexive transitive antisymmetric )
Poset) ) ) : ( ( ) ( )
Element of the
carrier of
(subrelstr b3 : ( ( non empty ) ( non empty ) strict_chain of b2 : ( ( auxiliary(i) auxiliary(ii) ) ( Relation-like the carrier of b1 : ( ( non empty reflexive transitive antisymmetric ) ( non empty reflexive transitive antisymmetric ) Poset) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty reflexive transitive antisymmetric ) ( non empty reflexive transitive antisymmetric ) Poset) : ( ( ) ( non empty ) set ) -valued auxiliary(i) auxiliary(ii) ) Relation of ) ) ) : ( (
strict V134(
b1 : ( ( non
empty reflexive transitive antisymmetric ) ( non
empty reflexive transitive antisymmetric )
Poset) ) ) ( non
empty strict reflexive transitive antisymmetric V134(
b1 : ( ( non
empty reflexive transitive antisymmetric ) ( non
empty reflexive transitive antisymmetric )
Poset) ) )
SubRelStr of
b1 : ( ( non
empty reflexive transitive antisymmetric ) ( non
empty reflexive transitive antisymmetric )
Poset) ) : ( ( ) ( non
empty )
set ) )
= "/\" (
((uparrow ("\/" (X : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ,L : ( ( non empty reflexive transitive antisymmetric ) ( non empty reflexive transitive antisymmetric ) Poset) )) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty reflexive transitive antisymmetric ) ( non empty reflexive transitive antisymmetric ) Poset) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty filtered upper ) Element of bool the carrier of b1 : ( ( non empty reflexive transitive antisymmetric ) ( non empty reflexive transitive antisymmetric ) Poset) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ C : ( ( non empty ) ( non empty ) strict_chain of b2 : ( ( auxiliary(i) auxiliary(ii) ) ( Relation-like the carrier of b1 : ( ( non empty reflexive transitive antisymmetric ) ( non empty reflexive transitive antisymmetric ) Poset) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty reflexive transitive antisymmetric ) ( non empty reflexive transitive antisymmetric ) Poset) : ( ( ) ( non empty ) set ) -valued auxiliary(i) auxiliary(ii) ) Relation of ) ) ) : ( ( ) ( )
Element of
bool the
carrier of
b1 : ( ( non
empty reflexive transitive antisymmetric ) ( non
empty reflexive transitive antisymmetric )
Poset) : ( ( ) ( non
empty )
set ) : ( ( ) ( non
empty )
set ) ) ,
L : ( ( non
empty reflexive transitive antisymmetric ) ( non
empty reflexive transitive antisymmetric )
Poset) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty reflexive transitive antisymmetric ) ( non
empty reflexive transitive antisymmetric )
Poset) : ( ( ) ( non
empty )
set ) ) & ( not
"\/" (
X : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ,
L : ( ( non
empty reflexive transitive antisymmetric ) ( non
empty reflexive transitive antisymmetric )
Poset) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty reflexive transitive antisymmetric ) ( non
empty reflexive transitive antisymmetric )
Poset) : ( ( ) ( non
empty )
set ) )
in C : ( ( non
empty ) ( non
empty )
strict_chain of
b2 : ( (
auxiliary(i) auxiliary(ii) ) (
Relation-like the
carrier of
b1 : ( ( non
empty reflexive transitive antisymmetric ) ( non
empty reflexive transitive antisymmetric )
Poset) : ( ( ) ( non
empty )
set )
-defined the
carrier of
b1 : ( ( non
empty reflexive transitive antisymmetric ) ( non
empty reflexive transitive antisymmetric )
Poset) : ( ( ) ( non
empty )
set )
-valued auxiliary(i) auxiliary(ii) )
Relation of ) ) implies
"\/" (
X : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ,
L : ( ( non
empty reflexive transitive antisymmetric ) ( non
empty reflexive transitive antisymmetric )
Poset) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty reflexive transitive antisymmetric ) ( non
empty reflexive transitive antisymmetric )
Poset) : ( ( ) ( non
empty )
set ) )
< "/\" (
((uparrow ("\/" (X : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ,L : ( ( non empty reflexive transitive antisymmetric ) ( non empty reflexive transitive antisymmetric ) Poset) )) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty reflexive transitive antisymmetric ) ( non empty reflexive transitive antisymmetric ) Poset) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty filtered upper ) Element of bool the carrier of b1 : ( ( non empty reflexive transitive antisymmetric ) ( non empty reflexive transitive antisymmetric ) Poset) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ C : ( ( non empty ) ( non empty ) strict_chain of b2 : ( ( auxiliary(i) auxiliary(ii) ) ( Relation-like the carrier of b1 : ( ( non empty reflexive transitive antisymmetric ) ( non empty reflexive transitive antisymmetric ) Poset) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty reflexive transitive antisymmetric ) ( non empty reflexive transitive antisymmetric ) Poset) : ( ( ) ( non empty ) set ) -valued auxiliary(i) auxiliary(ii) ) Relation of ) ) ) : ( ( ) ( )
Element of
bool the
carrier of
b1 : ( ( non
empty reflexive transitive antisymmetric ) ( non
empty reflexive transitive antisymmetric )
Poset) : ( ( ) ( non
empty )
set ) : ( ( ) ( non
empty )
set ) ) ,
L : ( ( non
empty reflexive transitive antisymmetric ) ( non
empty reflexive transitive antisymmetric )
Poset) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty reflexive transitive antisymmetric ) ( non
empty reflexive transitive antisymmetric )
Poset) : ( ( ) ( non
empty )
set ) ) ) ) ;
definition
let L be ( ( ) ( )
RelStr ) ;
let C be ( ( ) ( )
set ) ;
let R be ( ( ) (
Relation-like the
carrier of
L : ( ( ) ( )
RelStr ) : ( ( ) ( )
set )
-defined the
carrier of
L : ( ( ) ( )
RelStr ) : ( ( ) ( )
set )
-valued )
Relation of ) ;
pred R satisfies_SIC_on C means
for
x,
z being ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) st
x : ( ( ) ( )
Element of ( ( ) ( )
set ) )
in C : ( ( ) (
Relation-like the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-valued )
Element of
bool [: the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non
empty )
set ) : ( ( ) ( non
empty )
set ) ) &
z : ( ( ) ( )
Element of ( ( ) ( )
set ) )
in C : ( ( ) (
Relation-like the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-valued )
Element of
bool [: the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non
empty )
set ) : ( ( ) ( non
empty )
set ) ) &
[x : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,z : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ] : ( ( ) ( non
empty )
set )
in R : ( ( ) ( )
strict_chain of
C : ( ( ) (
Relation-like the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-valued )
Element of
bool [: the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non
empty )
set ) : ( ( ) ( non
empty )
set ) ) ) &
x : ( ( ) ( )
Element of ( ( ) ( )
set ) )
<> z : ( ( ) ( )
Element of ( ( ) ( )
set ) ) holds
ex
y being ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) st
(
y : ( ( ) ( )
Element of ( ( ) ( )
set ) )
in C : ( ( ) (
Relation-like the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-valued )
Element of
bool [: the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non
empty )
set ) : ( ( ) ( non
empty )
set ) ) &
[x : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,y : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ] : ( ( ) ( non
empty )
set )
in R : ( ( ) ( )
strict_chain of
C : ( ( ) (
Relation-like the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-valued )
Element of
bool [: the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non
empty )
set ) : ( ( ) ( non
empty )
set ) ) ) &
[y : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,z : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ] : ( ( ) ( non
empty )
set )
in R : ( ( ) ( )
strict_chain of
C : ( ( ) (
Relation-like the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
1-sorted ) : ( ( ) ( non
empty )
set )
-valued )
Element of
bool [: the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non
empty )
set ) : ( ( ) ( non
empty )
set ) ) ) &
x : ( ( ) ( )
Element of ( ( ) ( )
set ) )
<> y : ( ( ) ( )
Element of ( ( ) ( )
set ) ) );
end;
definition
let L be ( ( ) ( )
RelStr ) ;
let C be ( ( ) ( )
Subset of ) ;
end;
definition
let L be ( ( non
empty ) ( non
empty )
RelStr ) ;
let R be ( ( ) (
Relation-like the
carrier of
L : ( ( non
empty ) ( non
empty )
RelStr ) : ( ( ) ( non
empty )
set )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
RelStr ) : ( ( ) ( non
empty )
set )
-valued )
Relation of ) ;
let C be ( ( ) ( )
set ) ;
func SupBelow (
R,
C)
-> ( ( ) ( )
set )
means
for
y being ( ( ) ( )
set ) holds
(
y : ( ( ) ( )
set )
in it : ( (
Function-like V31(
R : ( ( ) (
Relation-like the
carrier of
L : ( ( non
empty ) ( non
empty )
RelStr ) : ( ( ) ( non
empty )
set )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
RelStr ) : ( ( ) ( non
empty )
set )
-valued )
Element of
bool [: the carrier of L : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non
empty )
set ) : ( ( ) ( non
empty )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
RelStr ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like R : ( ( ) (
Relation-like the
carrier of
L : ( ( non
empty ) ( non
empty )
RelStr ) : ( ( ) ( non
empty )
set )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
RelStr ) : ( ( ) ( non
empty )
set )
-valued )
Element of
bool [: the carrier of L : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non
empty )
set ) : ( ( ) ( non
empty )
set ) )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
RelStr ) : ( ( ) ( non
empty )
set )
-valued Function-like V31(
R : ( ( ) (
Relation-like the
carrier of
L : ( ( non
empty ) ( non
empty )
RelStr ) : ( ( ) ( non
empty )
set )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
RelStr ) : ( ( ) ( non
empty )
set )
-valued )
Element of
bool [: the carrier of L : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non
empty )
set ) : ( ( ) ( non
empty )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
RelStr ) : ( ( ) ( non
empty )
set ) ) )
Element of
bool [:R : ( ( ) ( Relation-like the carrier of L : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) -defined the carrier of L : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) -valued ) Element of bool [: the carrier of L : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , the carrier of L : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( )
set ) : ( ( ) ( non
empty )
set ) ) iff
y : ( ( ) ( )
set )
= sup (SetBelow (R : ( ( ) ( Relation-like the carrier of L : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) -defined the carrier of L : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) -valued ) Element of bool [: the carrier of L : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,C : ( ( ) ( ) strict_chain of R : ( ( ) ( Relation-like the carrier of L : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) -defined the carrier of L : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) -valued ) Element of bool [: the carrier of L : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ,y : ( ( ) ( ) set ) )) : ( ( ) ( )
Subset of ) : ( ( ) ( )
Element of the
carrier of
L : ( ( non
empty ) ( non
empty )
RelStr ) : ( ( ) ( non
empty )
set ) ) );
end;