:: CONLAT_1 semantic presentation

begin

definition
let C be ( ( ) ( ) 2-sorted ) ;
attr C is quasi-empty means :: CONLAT_1:def 1
( the carrier of C : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) is empty or the carrier' of C : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) is empty );
end;

registration
cluster non empty strict non void for ( ( ) ( ) 2-sorted ) ;
cluster strict non quasi-empty for ( ( ) ( ) 2-sorted ) ;
end;

registration
cluster empty strict void quasi-empty for ( ( ) ( ) 2-sorted ) ;
end;

definition
attr c1 is strict ;
struct ContextStr -> ( ( ) ( ) 2-sorted ) ;
aggr ContextStr(# carrier, carrier', Information #) -> ( ( strict ) ( strict ) ContextStr ) ;
sel Information c1 -> ( ( ) ( Relation-like the carrier of c1 : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -defined the carrier' of c1 : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -valued ) Relation of ,) ;
end;

registration
cluster non empty strict for ( ( ) ( ) ContextStr ) ;
cluster non quasi-empty strict for ( ( ) ( ) ContextStr ) ;
end;

definition
mode FormalContext is ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ;
end;

definition
let C be ( ( ) ( ) 2-sorted ) ;
mode Object of C is ( ( ) ( ) Element of ( ( ) ( ) set ) ) ;
mode Attribute of C is ( ( ) ( ) Element of the carrier' of C : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ;
end;

registration
let C be ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) ;
cluster the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( ) set ) -> non empty ;
cluster the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( ) set ) -> non empty ;
end;

registration
let C be ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) ;
cluster non empty for ( ( ) ( ) Element of bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;
cluster non empty for ( ( ) ( ) Element of bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

definition
let C be ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ;
let o be ( ( ) ( ) Object of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ;
let a be ( ( ) ( ) Attribute of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ;
pred o is-connected-with a means :: CONLAT_1:def 2
[o : ( ( ) ( ) set ) ,a : ( ( ) ( Relation-like o : ( ( ) ( ) set ) -defined o : ( ( ) ( ) set ) -valued ) Element of bool [:o : ( ( ) ( ) set ) ,o : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V15() ) set ) in the Information of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( Relation-like the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) -defined the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) -valued ) Relation of ,) ;
end;

notation
let C be ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ;
let o be ( ( ) ( ) Object of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ;
let a be ( ( ) ( ) Attribute of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ;
antonym o is-not-connected-with a for o is-connected-with a;
end;

begin

definition
let C be ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ;
func ObjectDerivation C -> ( ( Function-like V18( bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) means :: CONLAT_1:def 3
 for O being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds it : ( ( ) ( ) set ) . O : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) = { a : ( ( ) ( ) Attribute of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) where a is ( ( ) ( ) Attribute of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) ) : for o being ( ( ) ( ) Object of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) ) st o : ( ( ) ( ) Object of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) in O : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds
o : ( ( ) ( ) Object of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) is-connected-with a : ( ( ) ( ) Attribute of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) } ;
end;

definition
let C be ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ;
func AttributeDerivation C -> ( ( Function-like V18( bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) means :: CONLAT_1:def 4
 for A being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds it : ( ( ) ( ) set ) . A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) = { o : ( ( ) ( ) Object of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) where o is ( ( ) ( ) Object of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) ) : for a being ( ( ) ( ) Attribute of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) ) st a : ( ( ) ( ) Attribute of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) in A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds
o : ( ( ) ( ) Object of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) is-connected-with a : ( ( ) ( ) Attribute of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) } ;
end;

theorem :: CONLAT_1:1
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for o being ( ( ) ( ) Object of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) holds (ObjectDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . {o : ( ( ) ( ) Object of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) } : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) = { a : ( ( ) ( ) Attribute of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) where a is ( ( ) ( ) Attribute of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : o : ( ( ) ( ) Object of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) is-connected-with a : ( ( ) ( ) Attribute of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) } ;

theorem :: CONLAT_1:2
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for a being ( ( ) ( ) Attribute of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) holds (AttributeDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . {a : ( ( ) ( ) Attribute of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) } : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) = { o : ( ( ) ( ) Object of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) where o is ( ( ) ( ) Object of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : o : ( ( ) ( ) Object of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) is-connected-with a : ( ( ) ( ) Attribute of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) } ;

theorem :: CONLAT_1:3
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for O1, O2 being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) st O1 : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) c= O2 : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds
(ObjectDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . O2 : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) c= (ObjectDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . O1 : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: CONLAT_1:4
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for A1, A2 being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) st A1 : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) c= A2 : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds
(AttributeDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . A2 : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) c= (AttributeDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . A1 : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: CONLAT_1:5
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for O being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds O : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) c= (AttributeDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . ((ObjectDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . O : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: CONLAT_1:6
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for A being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) c= (ObjectDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . ((AttributeDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: CONLAT_1:7
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for O being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds (ObjectDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . O : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) = (ObjectDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . ((AttributeDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . ((ObjectDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . O : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: CONLAT_1:8
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for A being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds (AttributeDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) = (AttributeDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . ((ObjectDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . ((AttributeDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: CONLAT_1:9
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for O being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) )
for A being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds
( O : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) c= (AttributeDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) iff [:O : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ,A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( Relation-like the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) -defined the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) -valued ) Element of bool [: the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) , the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) c= the Information of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( Relation-like the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) -defined the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) -valued ) Relation of ,) ) ;

theorem :: CONLAT_1:10
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for O being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) )
for A being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds
( A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) c= (ObjectDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . O : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) iff [:O : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ,A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( Relation-like the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) -defined the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) -valued ) Element of bool [: the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) , the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) c= the Information of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( Relation-like the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) -defined the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) -valued ) Relation of ,) ) ;

theorem :: CONLAT_1:11
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for O being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) )
for A being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds
( O : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) c= (AttributeDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) iff A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) c= (ObjectDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . O : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ;

definition
let C be ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ;
func phi C -> ( ( Function-like V18( the carrier of (BoolePoset the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict V59() V60() V61() complete ) RelStr ) : ( ( ) ( non empty ) set ) , the carrier of (BoolePoset the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict V59() V60() V61() complete ) RelStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of (BoolePoset the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict V59() V60() V61() complete ) RelStr ) : ( ( ) ( non empty ) set ) -defined the carrier of (BoolePoset the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict V59() V60() V61() complete ) RelStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of (BoolePoset the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict V59() V60() V61() complete ) RelStr ) : ( ( ) ( non empty ) set ) ) V18( the carrier of (BoolePoset the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict V59() V60() V61() complete ) RelStr ) : ( ( ) ( non empty ) set ) , the carrier of (BoolePoset the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict V59() V60() V61() complete ) RelStr ) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) equals :: CONLAT_1:def 5
 ObjectDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( Function-like V18( bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ;
end;

definition
let C be ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ;
func psi C -> ( ( Function-like V18( the carrier of (BoolePoset the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict V59() V60() V61() complete ) RelStr ) : ( ( ) ( non empty ) set ) , the carrier of (BoolePoset the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict V59() V60() V61() complete ) RelStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of (BoolePoset the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict V59() V60() V61() complete ) RelStr ) : ( ( ) ( non empty ) set ) -defined the carrier of (BoolePoset the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict V59() V60() V61() complete ) RelStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of (BoolePoset the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict V59() V60() V61() complete ) RelStr ) : ( ( ) ( non empty ) set ) ) V18( the carrier of (BoolePoset the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict V59() V60() V61() complete ) RelStr ) : ( ( ) ( non empty ) set ) , the carrier of (BoolePoset the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict V59() V60() V61() complete ) RelStr ) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) equals :: CONLAT_1:def 6
 AttributeDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( Function-like V18( bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ;
end;

definition
let P, R be ( ( non empty ) ( non empty ) RelStr ) ;
let Con be ( ( ) ( ) Connection of P : ( ( non empty ) ( non empty ) RelStr ) ,R : ( ( non empty ) ( non empty ) RelStr ) ) ;
attr Con is co-Galois means :: CONLAT_1:def 7
 ex f being ( ( Function-like V18( the carrier of P : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) , the carrier of R : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ( Relation-like the carrier of P : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) -defined the carrier of R : ( ( ) ( ) set ) : ( ( ) ( ) set ) -valued Function-like non empty V14( the carrier of P : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) V18( the carrier of P : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) , the carrier of R : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) ex g being ( ( Function-like V18( the carrier of R : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of P : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of R : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined the carrier of P : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of R : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) V18( the carrier of R : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of P : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty ) set ) ) st
( Con : ( ( ) ( Relation-like R : ( ( ) ( ) set ) -defined R : ( ( ) ( ) set ) -valued ) Element of bool [:R : ( ( ) ( ) set ) ,R : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) = [f : ( ( Function-like V18( the carrier of P : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) , the carrier of R : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of P : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) -defined the carrier of R : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of P : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) ) V18( the carrier of P : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) , the carrier of R : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ,g : ( ( Function-like V18( the carrier of R : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) , the carrier of P : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of R : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) -defined the carrier of P : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of R : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) ) V18( the carrier of R : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) , the carrier of P : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V15() ) Connection of P : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) ,R : ( ( ) ( ) set ) ) & f : ( ( Function-like V18( the carrier of P : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) , the carrier of R : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of P : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) -defined the carrier of R : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of P : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) ) V18( the carrier of P : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) , the carrier of R : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is antitone & g : ( ( Function-like V18( the carrier of R : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) , the carrier of P : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of R : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) -defined the carrier of P : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of R : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) ) V18( the carrier of R : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) , the carrier of P : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is antitone & ( for p1, p2 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) )
for r1, r2 being ( ( ) ( ) Element of ( ( ) ( ) set ) ) holds
( p1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <= g : ( ( Function-like V18( the carrier of R : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) , the carrier of P : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of R : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) -defined the carrier of P : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of R : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) ) V18( the carrier of R : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) , the carrier of P : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . (f : ( ( Function-like V18( the carrier of P : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) , the carrier of R : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of P : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) -defined the carrier of R : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of P : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) ) V18( the carrier of P : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) , the carrier of R : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . p1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of R : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of P : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) & r1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <= f : ( ( Function-like V18( the carrier of P : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) , the carrier of R : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of P : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) -defined the carrier of R : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of P : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) ) V18( the carrier of P : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) , the carrier of R : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . (g : ( ( Function-like V18( the carrier of R : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) , the carrier of P : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of R : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) -defined the carrier of P : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of R : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) ) V18( the carrier of R : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) , the carrier of P : ( ( non empty ) ( non empty ) RelStr ) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . r1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of P : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of R : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ) );
end;

theorem :: CONLAT_1:12
for P, R being ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset)
for Con being ( ( ) ( ) Connection of P : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) ,R : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) )
for f being ( ( Function-like V18( the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) V18( the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for g being ( ( Function-like V18( the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) V18( the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st Con : ( ( ) ( ) Connection of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) ,b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) ) = [f : ( ( Function-like V18( the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) V18( the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ,g : ( ( Function-like V18( the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) V18( the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V15() ) Connection of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) ,b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) ) holds
( Con : ( ( ) ( ) Connection of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) ,b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) ) is co-Galois iff for p being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) )
for r being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <= g : ( ( Function-like V18( the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) V18( the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) iff r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <= f : ( ( Function-like V18( the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) V18( the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: CONLAT_1:13
for P, R being ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset)
for Con being ( ( ) ( ) Connection of P : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) ,R : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) ) st Con : ( ( ) ( ) Connection of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) ,b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) ) is co-Galois holds
for f being ( ( Function-like V18( the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) V18( the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for g being ( ( Function-like V18( the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) V18( the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st Con : ( ( ) ( ) Connection of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) ,b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) ) = [f : ( ( Function-like V18( the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) V18( the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ,g : ( ( Function-like V18( the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) V18( the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V15() ) Connection of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) ,b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) ) holds
( f : ( ( Function-like V18( the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) V18( the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f : ( ( Function-like V18( the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) V18( the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * (g : ( ( Function-like V18( the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) V18( the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like V18( the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) V18( the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) V18( the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) Element of bool [: the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) V18( the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) Element of bool [: the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & g : ( ( Function-like V18( the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) V18( the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = g : ( ( Function-like V18( the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) V18( the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * (f : ( ( Function-like V18( the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) V18( the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * g : ( ( Function-like V18( the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) V18( the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) V18( the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) Element of bool [: the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) V18( the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) ) ) Element of bool [: the carrier of b2 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty V59() V60() V61() ) ( non empty V59() V60() V61() ) Poset) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: CONLAT_1:14
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) holds [(phi C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( the carrier of (BoolePoset the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict V59() V60() V61() complete ) RelStr ) : ( ( ) ( non empty ) set ) , the carrier of (BoolePoset the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict V59() V60() V61() complete ) RelStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of (BoolePoset the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict V59() V60() V61() complete ) RelStr ) : ( ( ) ( non empty ) set ) -defined the carrier of (BoolePoset the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict V59() V60() V61() complete ) RelStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of (BoolePoset the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict V59() V60() V61() complete ) RelStr ) : ( ( ) ( non empty ) set ) ) V18( the carrier of (BoolePoset the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict V59() V60() V61() complete ) RelStr ) : ( ( ) ( non empty ) set ) , the carrier of (BoolePoset the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict V59() V60() V61() complete ) RelStr ) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ,(psi C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( the carrier of (BoolePoset the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict V59() V60() V61() complete ) RelStr ) : ( ( ) ( non empty ) set ) , the carrier of (BoolePoset the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict V59() V60() V61() complete ) RelStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of (BoolePoset the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict V59() V60() V61() complete ) RelStr ) : ( ( ) ( non empty ) set ) -defined the carrier of (BoolePoset the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict V59() V60() V61() complete ) RelStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14( the carrier of (BoolePoset the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict V59() V60() V61() complete ) RelStr ) : ( ( ) ( non empty ) set ) ) V18( the carrier of (BoolePoset the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict V59() V60() V61() complete ) RelStr ) : ( ( ) ( non empty ) set ) , the carrier of (BoolePoset the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict V59() V60() V61() complete ) RelStr ) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V15() ) Connection of BoolePoset the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( strict ) ( non empty strict V59() V60() V61() complete ) RelStr ) , BoolePoset the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( strict ) ( non empty strict V59() V60() V61() complete ) RelStr ) ) is co-Galois ;

theorem :: CONLAT_1:15
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for O1, O2 being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds (ObjectDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . (O1 : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) \/ O2 : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) = ((ObjectDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . O1 : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) /\ ((ObjectDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . O2 : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: CONLAT_1:16
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for A1, A2 being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds (AttributeDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . (A1 : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) \/ A2 : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) = ((AttributeDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . A1 : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) /\ ((AttributeDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . A2 : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: CONLAT_1:17
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) holds (ObjectDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . {} : ( ( ) ( empty ) set ) : ( ( ) ( ) set ) = the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ;

theorem :: CONLAT_1:18
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) holds (AttributeDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . {} : ( ( ) ( empty ) set ) : ( ( ) ( ) set ) = the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ;

begin

definition
let C be ( ( ) ( ) 2-sorted ) ;
attr c2 is strict ;
struct ConceptStr over C -> ;
aggr ConceptStr(# Extent, Intent #) -> ( ( strict ) ( strict ) ConceptStr over C : ( ( ) ( ) 2-sorted ) ) ;
sel Extent c2 -> ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ;
sel Intent c2 -> ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ;
end;

definition
let C be ( ( ) ( ) 2-sorted ) ;
let CP be ( ( ) ( ) ConceptStr over C : ( ( ) ( ) 2-sorted ) ) ;
attr CP is empty means :: CONLAT_1:def 8
( the Extent of CP : ( ( ) ( ) set ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) is empty & the Intent of CP : ( ( ) ( ) set ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) is empty );
attr CP is quasi-empty means :: CONLAT_1:def 9
( the Extent of CP : ( ( ) ( ) set ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) is empty or the Intent of CP : ( ( ) ( ) set ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) is empty );
end;

registration
let C be ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) ;
cluster strict non empty for ( ( ) ( ) ConceptStr over C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) ) ;
cluster strict quasi-empty for ( ( ) ( ) ConceptStr over C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) ) ;
end;

registration
let C be ( ( empty void ) ( empty trivial V49() void V54( {} : ( ( ) ( empty ) set ) ) V55() trivial' ) 2-sorted ) ;
cluster -> empty for ( ( ) ( ) ConceptStr over C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) ) ;
end;

registration
let C be ( ( quasi-empty ) ( quasi-empty ) 2-sorted ) ;
cluster -> quasi-empty for ( ( ) ( ) ConceptStr over C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) ) ;
end;

definition
let C be ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ;
let CP be ( ( ) ( ) ConceptStr over C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ;
attr CP is concept-like means :: CONLAT_1:def 10
( (ObjectDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) ) : ( ( Function-like V18( bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . the Extent of CP : ( ( ) ( ) set ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) = the Intent of CP : ( ( ) ( ) set ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) & (AttributeDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) ) : ( ( Function-like V18( bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . the Intent of CP : ( ( ) ( ) set ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) 2-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) = the Extent of CP : ( ( ) ( ) set ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) );
end;

registration
let C be ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ;
cluster non empty concept-like for ( ( ) ( ) ConceptStr over C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ;
end;

definition
let C be ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ;
mode FormalConcept of C is ( ( non empty concept-like ) ( non empty concept-like ) ConceptStr over C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ;
end;

registration
let C be ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ;
cluster strict non empty concept-like for ( ( ) ( ) ConceptStr over C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ;
end;

theorem :: CONLAT_1:19
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for O being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds
( ConceptStr(# ((AttributeDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . ((ObjectDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . O : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ,((ObjectDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . O : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) #) : ( ( strict ) ( strict ) ConceptStr over b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) is ( ( non empty concept-like ) ( non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) & ( for O9 being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) )
for A9 being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) st ConceptStr(# O9 : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ,A9 : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) ConceptStr over b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) is ( ( non empty concept-like ) ( non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) & O : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) c= O9 : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds
(AttributeDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . ((ObjectDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . O : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) c= O9 : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: CONLAT_1:20
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for O being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds
( ex A being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) st ConceptStr(# O : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ,A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) ConceptStr over b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) is ( ( non empty concept-like ) ( non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) iff (AttributeDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . ((ObjectDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . O : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) = O : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) ;

theorem :: CONLAT_1:21
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for A being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds
( ConceptStr(# ((AttributeDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ,((ObjectDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . ((AttributeDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) #) : ( ( strict ) ( strict ) ConceptStr over b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) is ( ( non empty concept-like ) ( non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) & ( for O9 being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) )
for A9 being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) st ConceptStr(# O9 : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ,A9 : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) ConceptStr over b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) is ( ( non empty concept-like ) ( non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) & A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) c= A9 : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds
(ObjectDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . ((AttributeDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) c= A9 : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: CONLAT_1:22
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for A being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds
( ex O being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) st ConceptStr(# O : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ,A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) ConceptStr over b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) is ( ( non empty concept-like ) ( non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) iff (ObjectDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . ((AttributeDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) = A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) ;

definition
let C be ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ;
let CP be ( ( ) ( ) ConceptStr over C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ;
attr CP is universal means :: CONLAT_1:def 11
 the Extent of CP : ( ( ) ( ) set ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ;
end;

definition
let C be ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ;
let CP be ( ( ) ( ) ConceptStr over C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ;
attr CP is co-universal means :: CONLAT_1:def 12
 the Intent of CP : ( ( ) ( ) set ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ;
end;

registration
let C be ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ;
cluster strict non empty concept-like universal for ( ( ) ( ) ConceptStr over C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ;
cluster strict non empty concept-like co-universal for ( ( ) ( ) ConceptStr over C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ;
end;

definition
let C be ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ;
func Concept-with-all-Objects C -> ( ( strict non empty concept-like universal ) ( strict non empty concept-like universal ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) means :: CONLAT_1:def 13
 ex O being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ex A being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) st
( it : ( ( ) ( ) set ) = ConceptStr(# O : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ,A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) ConceptStr over C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) & O : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = (AttributeDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( Function-like V18( bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . {} : ( ( ) ( empty ) set ) : ( ( ) ( ) set ) & A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = (ObjectDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( Function-like V18( bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . ((AttributeDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( Function-like V18( bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . {} : ( ( ) ( empty ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) );
end;

definition
let C be ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ;
func Concept-with-all-Attributes C -> ( ( strict non empty concept-like co-universal ) ( strict non empty concept-like co-universal ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) means :: CONLAT_1:def 14
 ex O being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ex A being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) st
( it : ( ( ) ( ) set ) = ConceptStr(# O : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ,A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) ConceptStr over C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) & O : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = (AttributeDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( Function-like V18( bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . ((ObjectDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( Function-like V18( bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . {} : ( ( ) ( empty ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) & A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = (ObjectDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( Function-like V18( bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . {} : ( ( ) ( empty ) set ) : ( ( ) ( ) set ) );
end;

theorem :: CONLAT_1:23
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) holds
( the Extent of (Concept-with-all-Objects C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( strict non empty concept-like universal ) ( strict non empty concept-like universal ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) & the Intent of (Concept-with-all-Attributes C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( strict non empty concept-like co-universal ) ( strict non empty concept-like co-universal ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) set ) ) ;

theorem :: CONLAT_1:24
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for CP being ( ( non empty concept-like ) ( non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) holds
( ( the Extent of CP : ( ( non empty concept-like ) ( non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = {} : ( ( ) ( empty ) set ) implies CP : ( ( non empty concept-like ) ( non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) is co-universal ) & ( the Intent of CP : ( ( non empty concept-like ) ( non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = {} : ( ( ) ( empty ) set ) implies CP : ( ( non empty concept-like ) ( non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) is universal ) ) ;

theorem :: CONLAT_1:25
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for CP being ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) holds
( ( the Extent of CP : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = {} : ( ( ) ( empty ) set ) implies CP : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) = Concept-with-all-Attributes C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( strict non empty concept-like co-universal ) ( strict non empty concept-like co-universal ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) & ( the Intent of CP : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = {} : ( ( ) ( empty ) set ) implies CP : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) = Concept-with-all-Objects C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( strict non empty concept-like universal ) ( strict non empty concept-like universal ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) ;

theorem :: CONLAT_1:26
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for CP being ( ( quasi-empty ) ( quasi-empty ) ConceptStr over C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) holds
( not CP : ( ( quasi-empty ) ( quasi-empty ) ConceptStr over b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) is ( ( non empty concept-like ) ( non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) or CP : ( ( quasi-empty ) ( quasi-empty ) ConceptStr over b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) is universal or CP : ( ( quasi-empty ) ( quasi-empty ) ConceptStr over b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) is co-universal ) ;

theorem :: CONLAT_1:27
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for CP being ( ( quasi-empty ) ( quasi-empty ) ConceptStr over C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) holds
( not CP : ( ( quasi-empty ) ( quasi-empty ) ConceptStr over b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) is ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) or CP : ( ( quasi-empty ) ( quasi-empty ) ConceptStr over b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) = Concept-with-all-Objects C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( strict non empty concept-like universal ) ( strict non empty concept-like universal ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) or CP : ( ( quasi-empty ) ( quasi-empty ) ConceptStr over b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) = Concept-with-all-Attributes C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( strict non empty concept-like co-universal ) ( strict non empty concept-like co-universal ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ;

definition
let C be ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ;
mode Set-of-FormalConcepts of C -> ( ( non empty ) ( non empty ) set ) means :: CONLAT_1:def 15
 for X being ( ( ) ( ) set ) st X : ( ( ) ( ) set ) in it : ( ( ) ( ) set ) holds
X : ( ( ) ( ) set ) is ( ( non empty concept-like ) ( non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ;
end;

definition
let C be ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ;
let FCS be ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ;
:: original: Element
redefine mode Element of FCS -> ( ( non empty concept-like ) ( non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ;
end;

definition
let C be ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ;
let CP1, CP2 be ( ( non empty concept-like ) ( non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ;
pred CP1 is-SubConcept-of CP2 means :: CONLAT_1:def 16
 the Extent of CP1 : ( ( ) ( ) set ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) c= the Extent of CP2 : ( ( ) ( Relation-like CP1 : ( ( ) ( ) set ) -defined CP1 : ( ( ) ( ) set ) -valued ) Element of bool [:CP1 : ( ( ) ( ) set ) ,CP1 : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ;
end;

notation
let C be ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ;
let CP1, CP2 be ( ( non empty concept-like ) ( non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ;
synonym CP2 is-SuperConcept-of CP1 for CP1 is-SubConcept-of CP2;
end;

theorem :: CONLAT_1:28
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for CP1, CP2 being ( ( non empty concept-like ) ( non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) holds
( CP1 : ( ( non empty concept-like ) ( non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) is-SubConcept-of CP2 : ( ( non empty concept-like ) ( non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) iff the Intent of CP2 : ( ( non empty concept-like ) ( non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) c= the Intent of CP1 : ( ( non empty concept-like ) ( non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) ;

theorem :: CONLAT_1:29
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for CP1, CP2 being ( ( non empty concept-like ) ( non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) holds
( CP1 : ( ( non empty concept-like ) ( non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) is-SuperConcept-of CP2 : ( ( non empty concept-like ) ( non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) iff the Intent of CP1 : ( ( non empty concept-like ) ( non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) c= the Intent of CP2 : ( ( non empty concept-like ) ( non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) ;

theorem :: CONLAT_1:30
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for CP being ( ( non empty concept-like ) ( non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) holds
( CP : ( ( non empty concept-like ) ( non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) is-SubConcept-of Concept-with-all-Objects C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( strict non empty concept-like universal ) ( strict non empty concept-like universal ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) & Concept-with-all-Attributes C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( strict non empty concept-like co-universal ) ( strict non empty concept-like co-universal ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) is-SubConcept-of CP : ( ( non empty concept-like ) ( non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ;

begin

definition
let C be ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ;
func B-carrier C -> ( ( non empty ) ( non empty ) set ) equals :: CONLAT_1:def 17
 { ConceptStr(# E : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ,I : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) ConceptStr over C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) where E is ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) , I is ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) : ( not ConceptStr(# E : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ,I : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) ConceptStr over C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) is empty & (ObjectDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( Function-like V18( bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . E : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) = I : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) & (AttributeDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( Function-like V18( bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . I : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) = E : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) } ;
end;

definition
let C be ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ;
:: original: B-carrier
redefine func B-carrier C -> ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ;
end;

registration
let C be ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ;
cluster B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( non empty ) ( non empty ) set ) -> non empty ;
end;

theorem :: CONLAT_1:31
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for CP being ( ( ) ( ) set ) holds
( CP : ( ( ) ( ) set ) in B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) iff CP : ( ( ) ( ) set ) is ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ;

definition
let C be ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ;
func B-meet C -> ( ( Function-like V18([:(B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ,(B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) :] : ( ( ) ( non empty ) set ) , B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ) ) ( Relation-like [:(B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ,(B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) :] : ( ( ) ( non empty ) set ) -defined B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) -valued Function-like non empty V14([:(B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ,(B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) :] : ( ( ) ( non empty ) set ) ) V18([:(B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ,(B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) :] : ( ( ) ( non empty ) set ) , B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ) ) BinOp of B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ) means :: CONLAT_1:def 18
 for CP1, CP2 being ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ex O being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ex A being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) st
( it : ( ( ) ( ) set ) . (CP1 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,CP2 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) : ( ( ) ( ) set ) = ConceptStr(# O : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ,A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) ConceptStr over C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) & O : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = the Extent of CP1 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) /\ the Extent of CP2 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = (ObjectDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( Function-like V18( bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . ((AttributeDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( Function-like V18( bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . ( the Intent of CP1 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) \/ the Intent of CP2 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) );
end;

definition
let C be ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ;
func B-join C -> ( ( Function-like V18([:(B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ,(B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) :] : ( ( ) ( non empty ) set ) , B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ) ) ( Relation-like [:(B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ,(B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) :] : ( ( ) ( non empty ) set ) -defined B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) -valued Function-like non empty V14([:(B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ,(B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) :] : ( ( ) ( non empty ) set ) ) V18([:(B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ,(B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) :] : ( ( ) ( non empty ) set ) , B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ) ) BinOp of B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ) means :: CONLAT_1:def 19
 for CP1, CP2 being ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ex O being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ex A being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) st
( it : ( ( ) ( ) set ) . (CP1 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,CP2 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) : ( ( ) ( ) set ) = ConceptStr(# O : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ,A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) ConceptStr over C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) & O : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = (AttributeDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( Function-like V18( bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . ((ObjectDerivation C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( Function-like V18( bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V14( bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V18( bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . ( the Extent of CP1 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) \/ the Extent of CP2 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool the carrier of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) & A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = the Intent of CP1 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) /\ the Intent of CP2 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool the carrier' of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) );
end;

theorem :: CONLAT_1:32
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for CP1, CP2 being ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) holds (B-meet C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) ( Relation-like [:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) -defined B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) -valued Function-like non empty V14([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) ) V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) BinOp of B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) . (CP1 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,CP2 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) : ( ( ) ( ) set ) = (B-meet C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) ( Relation-like [:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) -defined B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) -valued Function-like non empty V14([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) ) V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) BinOp of B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) . (CP2 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,CP1 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) : ( ( ) ( ) set ) ;

theorem :: CONLAT_1:33
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for CP1, CP2 being ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) holds (B-join C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) ( Relation-like [:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) -defined B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) -valued Function-like non empty V14([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) ) V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) BinOp of B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) . (CP1 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,CP2 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) : ( ( ) ( ) set ) = (B-join C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) ( Relation-like [:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) -defined B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) -valued Function-like non empty V14([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) ) V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) BinOp of B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) . (CP2 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,CP1 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) : ( ( ) ( ) set ) ;

theorem :: CONLAT_1:34
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for CP1, CP2, CP3 being ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) holds (B-meet C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) ( Relation-like [:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) -defined B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) -valued Function-like non empty V14([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) ) V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) BinOp of B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) . (CP1 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,((B-meet C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) ( Relation-like [:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) -defined B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) -valued Function-like non empty V14([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) ) V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) BinOp of B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) . (CP2 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,CP3 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = (B-meet C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) ( Relation-like [:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) -defined B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) -valued Function-like non empty V14([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) ) V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) BinOp of B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) . (((B-meet C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) ( Relation-like [:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) -defined B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) -valued Function-like non empty V14([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) ) V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) BinOp of B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) . (CP1 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,CP2 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) )) : ( ( ) ( ) set ) ,CP3 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) : ( ( ) ( ) set ) ;

theorem :: CONLAT_1:35
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for CP1, CP2, CP3 being ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) holds (B-join C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) ( Relation-like [:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) -defined B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) -valued Function-like non empty V14([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) ) V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) BinOp of B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) . (CP1 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,((B-join C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) ( Relation-like [:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) -defined B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) -valued Function-like non empty V14([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) ) V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) BinOp of B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) . (CP2 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,CP3 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = (B-join C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) ( Relation-like [:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) -defined B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) -valued Function-like non empty V14([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) ) V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) BinOp of B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) . (((B-join C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) ( Relation-like [:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) -defined B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) -valued Function-like non empty V14([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) ) V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) BinOp of B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) . (CP1 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,CP2 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) )) : ( ( ) ( ) set ) ,CP3 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) : ( ( ) ( ) set ) ;

theorem :: CONLAT_1:36
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for CP1, CP2 being ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) holds (B-join C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) ( Relation-like [:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) -defined B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) -valued Function-like non empty V14([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) ) V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) BinOp of B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) . (((B-meet C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) ( Relation-like [:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) -defined B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) -valued Function-like non empty V14([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) ) V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) BinOp of B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) . (CP1 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,CP2 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) )) : ( ( ) ( ) set ) ,CP2 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) : ( ( ) ( ) set ) = CP2 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ;

theorem :: CONLAT_1:37
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for CP1, CP2 being ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) holds (B-meet C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) ( Relation-like [:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) -defined B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) -valued Function-like non empty V14([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) ) V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) BinOp of B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) . (CP1 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,((B-join C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) ( Relation-like [:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) -defined B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) -valued Function-like non empty V14([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) ) V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) BinOp of B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) . (CP1 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,CP2 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = CP1 : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ;

theorem :: CONLAT_1:38
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for CP being ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) holds (B-meet C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) ( Relation-like [:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) -defined B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) -valued Function-like non empty V14([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) ) V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) BinOp of B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) . (CP : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(Concept-with-all-Objects C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( strict non empty concept-like universal ) ( strict non empty concept-like universal ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) : ( ( ) ( ) set ) = CP : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ;

theorem :: CONLAT_1:39
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for CP being ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) holds (B-join C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) ( Relation-like [:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) -defined B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) -valued Function-like non empty V14([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) ) V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) BinOp of B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) . (CP : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(Concept-with-all-Objects C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( strict non empty concept-like universal ) ( strict non empty concept-like universal ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) : ( ( ) ( ) set ) = Concept-with-all-Objects C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( strict non empty concept-like universal ) ( strict non empty concept-like universal ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ;

theorem :: CONLAT_1:40
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for CP being ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) holds (B-join C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) ( Relation-like [:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) -defined B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) -valued Function-like non empty V14([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) ) V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) BinOp of B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) . (CP : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(Concept-with-all-Attributes C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( strict non empty concept-like co-universal ) ( strict non empty concept-like co-universal ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) : ( ( ) ( ) set ) = CP : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ;

theorem :: CONLAT_1:41
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for CP being ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) holds (B-meet C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( Function-like V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) ( Relation-like [:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) -defined B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) -valued Function-like non empty V14([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) ) V18([:(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) :] : ( ( ) ( non empty ) set ) , B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ) BinOp of B-carrier b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( ) ( non empty ) Set-of-FormalConcepts of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) . (CP : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ,(Concept-with-all-Attributes C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) : ( ( strict non empty concept-like co-universal ) ( strict non empty concept-like co-universal ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) : ( ( ) ( ) set ) = Concept-with-all-Attributes C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( strict non empty concept-like co-universal ) ( strict non empty concept-like co-universal ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ;

definition
let C be ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ;
func ConceptLattice C -> ( ( non empty strict ) ( non empty strict ) LattStr ) equals :: CONLAT_1:def 20
 LattStr(# (B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ,(B-join C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( Function-like V18([:(B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ,(B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) :] : ( ( ) ( non empty ) set ) , B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ) ) ( Relation-like [:(B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ,(B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) :] : ( ( ) ( non empty ) set ) -defined B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) -valued Function-like non empty V14([:(B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ,(B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) :] : ( ( ) ( non empty ) set ) ) V18([:(B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ,(B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) :] : ( ( ) ( non empty ) set ) , B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ) ) BinOp of B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ) ,(B-meet C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( Function-like V18([:(B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ,(B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) :] : ( ( ) ( non empty ) set ) , B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ) ) ( Relation-like [:(B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ,(B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) :] : ( ( ) ( non empty ) set ) -defined B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) -valued Function-like non empty V14([:(B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ,(B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) :] : ( ( ) ( non empty ) set ) ) V18([:(B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ,(B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) :] : ( ( ) ( non empty ) set ) , B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ) ) BinOp of B-carrier C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( ) ( non empty ) Set-of-FormalConcepts of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ) #) : ( ( strict ) ( non empty strict ) LattStr ) ;
end;

theorem :: CONLAT_1:42
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) holds ConceptLattice C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( non empty strict ) ( non empty strict ) LattStr ) is ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ;

registration
let C be ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ;
cluster ConceptLattice C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( non empty strict ) ( non empty strict ) LattStr ) -> non empty strict Lattice-like ;
end;

definition
let C be ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ;
let S be ( ( non empty ) ( non empty ) Subset of ) ;
:: original: Element
redefine mode Element of S -> ( ( non empty concept-like ) ( non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) ;
end;

definition
let C be ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ;
let CP be ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;
func CP @ -> ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) ) equals :: CONLAT_1:def 21
CP : ( ( ) ( ) set ) ;
end;

theorem :: CONLAT_1:43
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext)
for CP1, CP2 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( CP1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= CP2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) iff CP1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) @ : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) is-SubConcept-of CP2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) @ : ( ( strict non empty concept-like ) ( strict non empty concept-like ) FormalConcept of b1 : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ) ) ;

theorem :: CONLAT_1:44
for C being ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) holds ConceptLattice C : ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) : ( ( non empty strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) is ( ( non empty Lattice-like complete ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like complete ) Lattice) ;

registration
let C be ( ( non quasi-empty ) ( non quasi-empty ) FormalContext) ;
cluster ConceptLattice C : ( ( non quasi-empty ) ( non quasi-empty ) ContextStr ) : ( ( non empty strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) -> non empty strict complete ;
end;