begin
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) )
for
h being ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
for
A9,
B9,
C9,
D9 being ( ( ) ( )
set ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
= (((B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) .--> B9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b4 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non empty ) set ) -defined bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -defined {b4 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) +* (C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) .--> C9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b5 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non empty ) set ) -defined bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -defined {b5 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) .--> D9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b6 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non empty ) set ) -defined bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -defined {b6 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( (
Relation-like Function-like ) (
Relation-like Function-like )
set )
+* (A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) .--> A9 : ( ( ) ( ) set ) ) : ( ( ) (
trivial Relation-like {b3 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set )
-defined bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non
empty )
set ) : ( ( ) ( non
empty )
set )
-defined {b3 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set )
-defined Function-like one-to-one )
set ) : ( (
Relation-like Function-like ) (
Relation-like Function-like )
set ) holds
rng h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function) : ( ( ) ( )
set )
= {(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) ,(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) ,(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) ,(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) )
for
z,
u being ( ( ) ( )
Element of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) is
independent &
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
EqClass (
z : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
(C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
b5 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' b6 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
= EqClass (
u : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
(C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
b5 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' b6 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) holds
EqClass (
u : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
(CompF (A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,G : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
CompF (
b3 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
b2 : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
meets EqClass (
z : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
(CompF (B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,G : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
CompF (
b4 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
b2 : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) )
for
z,
u being ( ( ) ( )
Element of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) is
independent &
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
EqClass (
z : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
b5 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
= EqClass (
u : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
b5 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) holds
EqClass (
u : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
(CompF (A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,G : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
CompF (
b3 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
b2 : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
meets EqClass (
z : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
(CompF (B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,G : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
CompF (
b4 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
b2 : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ;
theorem
for
A,
B,
C,
D,
E being ( ( ) ( )
set )
for
h being ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
for
A9,
B9,
C9,
D9,
E9 being ( ( ) ( )
set ) st
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
= ((((B : ( ( ) ( ) set ) .--> B9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b2 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) +* (C : ( ( ) ( ) set ) .--> C9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b3 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (D : ( ( ) ( ) set ) .--> D9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b4 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (E : ( ( ) ( ) set ) .--> E9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b5 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( (
Relation-like Function-like ) (
Relation-like Function-like )
set )
+* (A : ( ( ) ( ) set ) .--> A9 : ( ( ) ( ) set ) ) : ( ( ) (
trivial Relation-like {b1 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined Function-like one-to-one )
set ) : ( (
Relation-like Function-like ) (
Relation-like Function-like )
set ) holds
dom h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function) : ( ( ) ( )
set )
= {A : ( ( ) ( ) set ) ,B : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ,D : ( ( ) ( ) set ) ,E : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) ;
theorem
for
A,
B,
C,
D,
E being ( ( ) ( )
set )
for
h being ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
for
A9,
B9,
C9,
D9,
E9 being ( ( ) ( )
set ) st
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
= ((((B : ( ( ) ( ) set ) .--> B9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b2 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) +* (C : ( ( ) ( ) set ) .--> C9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b3 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (D : ( ( ) ( ) set ) .--> D9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b4 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (E : ( ( ) ( ) set ) .--> E9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b5 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( (
Relation-like Function-like ) (
Relation-like Function-like )
set )
+* (A : ( ( ) ( ) set ) .--> A9 : ( ( ) ( ) set ) ) : ( ( ) (
trivial Relation-like {b1 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined Function-like one-to-one )
set ) : ( (
Relation-like Function-like ) (
Relation-like Function-like )
set ) holds
rng h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function) : ( ( ) ( )
set )
= {(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . A : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . B : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . C : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . D : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . E : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) )
for
z,
u being ( ( ) ( )
Element of
Y : ( ( non
empty ) ( non
empty )
set ) )
for
h being ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) is
independent &
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
EqClass (
u : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
(((B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
((b4 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b5 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b6 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' b7 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
meets EqClass (
z : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
b3 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) )
for
z,
u being ( ( ) ( )
Element of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) is
independent &
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
EqClass (
z : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
((C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
(b5 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b6 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' b7 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
= EqClass (
u : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
((C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
(b5 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b6 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' b7 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) holds
EqClass (
u : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
(CompF (A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,G : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
CompF (
b3 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
b2 : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
meets EqClass (
z : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
(CompF (B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,G : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
CompF (
b4 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
b2 : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
CompF (
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
= (((B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
CompF (
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
= (((A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
CompF (
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
= (((A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
CompF (
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
= (((A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
CompF (
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
= (((A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
CompF (
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
= (((A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
A,
B,
C,
D,
E,
F being ( ( ) ( )
set )
for
h being ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
for
A9,
B9,
C9,
D9,
E9,
F9 being ( ( ) ( )
set ) st
A : ( ( ) ( )
set )
<> B : ( ( ) ( )
set ) &
A : ( ( ) ( )
set )
<> C : ( ( ) ( )
set ) &
A : ( ( ) ( )
set )
<> D : ( ( ) ( )
set ) &
A : ( ( ) ( )
set )
<> E : ( ( ) ( )
set ) &
A : ( ( ) ( )
set )
<> F : ( ( ) ( )
set ) &
B : ( ( ) ( )
set )
<> C : ( ( ) ( )
set ) &
B : ( ( ) ( )
set )
<> D : ( ( ) ( )
set ) &
B : ( ( ) ( )
set )
<> E : ( ( ) ( )
set ) &
B : ( ( ) ( )
set )
<> F : ( ( ) ( )
set ) &
C : ( ( ) ( )
set )
<> D : ( ( ) ( )
set ) &
C : ( ( ) ( )
set )
<> E : ( ( ) ( )
set ) &
C : ( ( ) ( )
set )
<> F : ( ( ) ( )
set ) &
D : ( ( ) ( )
set )
<> E : ( ( ) ( )
set ) &
D : ( ( ) ( )
set )
<> F : ( ( ) ( )
set ) &
E : ( ( ) ( )
set )
<> F : ( ( ) ( )
set ) &
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
= (((((B : ( ( ) ( ) set ) .--> B9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b2 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) +* (C : ( ( ) ( ) set ) .--> C9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b3 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (D : ( ( ) ( ) set ) .--> D9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b4 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (E : ( ( ) ( ) set ) .--> E9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b5 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (F : ( ( ) ( ) set ) .--> F9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b6 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( (
Relation-like Function-like ) (
Relation-like Function-like )
set )
+* (A : ( ( ) ( ) set ) .--> A9 : ( ( ) ( ) set ) ) : ( ( ) (
trivial Relation-like {b1 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined Function-like one-to-one )
set ) : ( (
Relation-like Function-like ) (
Relation-like Function-like )
set ) holds
(
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. A : ( ( ) ( )
set ) : ( ( ) ( )
set )
= A9 : ( ( ) ( )
set ) &
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. B : ( ( ) ( )
set ) : ( ( ) ( )
set )
= B9 : ( ( ) ( )
set ) &
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. C : ( ( ) ( )
set ) : ( ( ) ( )
set )
= C9 : ( ( ) ( )
set ) &
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. D : ( ( ) ( )
set ) : ( ( ) ( )
set )
= D9 : ( ( ) ( )
set ) &
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. E : ( ( ) ( )
set ) : ( ( ) ( )
set )
= E9 : ( ( ) ( )
set ) &
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. F : ( ( ) ( )
set ) : ( ( ) ( )
set )
= F9 : ( ( ) ( )
set ) ) ;
theorem
for
A,
B,
C,
D,
E,
F being ( ( ) ( )
set )
for
h being ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
for
A9,
B9,
C9,
D9,
E9,
F9 being ( ( ) ( )
set ) st
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
= (((((B : ( ( ) ( ) set ) .--> B9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b2 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) +* (C : ( ( ) ( ) set ) .--> C9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b3 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (D : ( ( ) ( ) set ) .--> D9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b4 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (E : ( ( ) ( ) set ) .--> E9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b5 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (F : ( ( ) ( ) set ) .--> F9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b6 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( (
Relation-like Function-like ) (
Relation-like Function-like )
set )
+* (A : ( ( ) ( ) set ) .--> A9 : ( ( ) ( ) set ) ) : ( ( ) (
trivial Relation-like {b1 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined Function-like one-to-one )
set ) : ( (
Relation-like Function-like ) (
Relation-like Function-like )
set ) holds
dom h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function) : ( ( ) ( )
set )
= {A : ( ( ) ( ) set ) ,B : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ,D : ( ( ) ( ) set ) ,E : ( ( ) ( ) set ) ,F : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) ;
theorem
for
A,
B,
C,
D,
E,
F being ( ( ) ( )
set )
for
h being ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
for
A9,
B9,
C9,
D9,
E9,
F9 being ( ( ) ( )
set ) st
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
= (((((B : ( ( ) ( ) set ) .--> B9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b2 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) +* (C : ( ( ) ( ) set ) .--> C9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b3 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (D : ( ( ) ( ) set ) .--> D9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b4 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (E : ( ( ) ( ) set ) .--> E9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b5 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (F : ( ( ) ( ) set ) .--> F9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b6 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( (
Relation-like Function-like ) (
Relation-like Function-like )
set )
+* (A : ( ( ) ( ) set ) .--> A9 : ( ( ) ( ) set ) ) : ( ( ) (
trivial Relation-like {b1 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined Function-like one-to-one )
set ) : ( (
Relation-like Function-like ) (
Relation-like Function-like )
set ) holds
rng h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function) : ( ( ) ( )
set )
= {(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . A : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . B : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . C : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . D : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . E : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . F : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) )
for
z,
u being ( ( ) ( )
Element of
Y : ( ( non
empty ) ( non
empty )
set ) )
for
h being ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) is
independent &
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
EqClass (
u : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
((((B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
(((b4 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b5 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b6 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b7 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' b8 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
meets EqClass (
z : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
b3 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) )
for
z,
u being ( ( ) ( )
Element of
Y : ( ( non
empty ) ( non
empty )
set ) )
for
h being ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) is
independent &
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
EqClass (
z : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
(((C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
((b5 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b6 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b7 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' b8 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
= EqClass (
u : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
(((C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
((b5 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b6 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b7 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' b8 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) holds
EqClass (
u : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
(CompF (A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,G : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
CompF (
b3 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
b2 : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
meets EqClass (
z : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
(CompF (B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,G : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
CompF (
b4 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
b2 : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ;
begin
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F,
J being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
CompF (
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
= ((((B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F,
J being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
CompF (
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
= ((((A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F,
J being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
CompF (
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
= ((((A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F,
J being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
CompF (
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
= ((((A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F,
J being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
CompF (
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
= ((((A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F,
J being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
CompF (
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
= ((((A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F,
J being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
CompF (
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
= ((((A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
A,
B,
C,
D,
E,
F,
J being ( ( ) ( )
set )
for
h being ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
for
A9,
B9,
C9,
D9,
E9,
F9,
J9 being ( ( ) ( )
set ) st
A : ( ( ) ( )
set )
<> B : ( ( ) ( )
set ) &
A : ( ( ) ( )
set )
<> C : ( ( ) ( )
set ) &
A : ( ( ) ( )
set )
<> D : ( ( ) ( )
set ) &
A : ( ( ) ( )
set )
<> E : ( ( ) ( )
set ) &
A : ( ( ) ( )
set )
<> F : ( ( ) ( )
set ) &
A : ( ( ) ( )
set )
<> J : ( ( ) ( )
set ) &
B : ( ( ) ( )
set )
<> C : ( ( ) ( )
set ) &
B : ( ( ) ( )
set )
<> D : ( ( ) ( )
set ) &
B : ( ( ) ( )
set )
<> E : ( ( ) ( )
set ) &
B : ( ( ) ( )
set )
<> F : ( ( ) ( )
set ) &
B : ( ( ) ( )
set )
<> J : ( ( ) ( )
set ) &
C : ( ( ) ( )
set )
<> D : ( ( ) ( )
set ) &
C : ( ( ) ( )
set )
<> E : ( ( ) ( )
set ) &
C : ( ( ) ( )
set )
<> F : ( ( ) ( )
set ) &
C : ( ( ) ( )
set )
<> J : ( ( ) ( )
set ) &
D : ( ( ) ( )
set )
<> E : ( ( ) ( )
set ) &
D : ( ( ) ( )
set )
<> F : ( ( ) ( )
set ) &
D : ( ( ) ( )
set )
<> J : ( ( ) ( )
set ) &
E : ( ( ) ( )
set )
<> F : ( ( ) ( )
set ) &
E : ( ( ) ( )
set )
<> J : ( ( ) ( )
set ) &
F : ( ( ) ( )
set )
<> J : ( ( ) ( )
set ) &
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
= ((((((B : ( ( ) ( ) set ) .--> B9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b2 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) +* (C : ( ( ) ( ) set ) .--> C9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b3 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (D : ( ( ) ( ) set ) .--> D9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b4 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (E : ( ( ) ( ) set ) .--> E9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b5 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (F : ( ( ) ( ) set ) .--> F9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b6 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (J : ( ( ) ( ) set ) .--> J9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b7 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( (
Relation-like Function-like ) (
Relation-like Function-like )
set )
+* (A : ( ( ) ( ) set ) .--> A9 : ( ( ) ( ) set ) ) : ( ( ) (
trivial Relation-like {b1 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined Function-like one-to-one )
set ) : ( (
Relation-like Function-like ) (
Relation-like Function-like )
set ) holds
(
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. A : ( ( ) ( )
set ) : ( ( ) ( )
set )
= A9 : ( ( ) ( )
set ) &
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. B : ( ( ) ( )
set ) : ( ( ) ( )
set )
= B9 : ( ( ) ( )
set ) &
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. C : ( ( ) ( )
set ) : ( ( ) ( )
set )
= C9 : ( ( ) ( )
set ) &
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. D : ( ( ) ( )
set ) : ( ( ) ( )
set )
= D9 : ( ( ) ( )
set ) &
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. E : ( ( ) ( )
set ) : ( ( ) ( )
set )
= E9 : ( ( ) ( )
set ) &
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. F : ( ( ) ( )
set ) : ( ( ) ( )
set )
= F9 : ( ( ) ( )
set ) &
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. J : ( ( ) ( )
set ) : ( ( ) ( )
set )
= J9 : ( ( ) ( )
set ) ) ;
theorem
for
A,
B,
C,
D,
E,
F,
J being ( ( ) ( )
set )
for
h being ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
for
A9,
B9,
C9,
D9,
E9,
F9,
J9 being ( ( ) ( )
set ) st
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
= ((((((B : ( ( ) ( ) set ) .--> B9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b2 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) +* (C : ( ( ) ( ) set ) .--> C9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b3 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (D : ( ( ) ( ) set ) .--> D9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b4 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (E : ( ( ) ( ) set ) .--> E9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b5 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (F : ( ( ) ( ) set ) .--> F9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b6 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (J : ( ( ) ( ) set ) .--> J9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b7 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( (
Relation-like Function-like ) (
Relation-like Function-like )
set )
+* (A : ( ( ) ( ) set ) .--> A9 : ( ( ) ( ) set ) ) : ( ( ) (
trivial Relation-like {b1 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined Function-like one-to-one )
set ) : ( (
Relation-like Function-like ) (
Relation-like Function-like )
set ) holds
dom h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function) : ( ( ) ( )
set )
= {A : ( ( ) ( ) set ) ,B : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ,D : ( ( ) ( ) set ) ,E : ( ( ) ( ) set ) ,F : ( ( ) ( ) set ) ,J : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) ;
theorem
for
A,
B,
C,
D,
E,
F,
J being ( ( ) ( )
set )
for
h being ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
for
A9,
B9,
C9,
D9,
E9,
F9,
J9 being ( ( ) ( )
set ) st
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
= ((((((B : ( ( ) ( ) set ) .--> B9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b2 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) +* (C : ( ( ) ( ) set ) .--> C9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b3 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (D : ( ( ) ( ) set ) .--> D9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b4 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (E : ( ( ) ( ) set ) .--> E9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b5 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (F : ( ( ) ( ) set ) .--> F9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b6 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (J : ( ( ) ( ) set ) .--> J9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b7 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( (
Relation-like Function-like ) (
Relation-like Function-like )
set )
+* (A : ( ( ) ( ) set ) .--> A9 : ( ( ) ( ) set ) ) : ( ( ) (
trivial Relation-like {b1 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined Function-like one-to-one )
set ) : ( (
Relation-like Function-like ) (
Relation-like Function-like )
set ) holds
rng h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function) : ( ( ) ( )
set )
= {(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . A : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . B : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . C : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . D : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . E : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . F : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . J : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F,
J being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) )
for
z,
u being ( ( ) ( )
Element of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) is
independent &
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
EqClass (
u : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
(((((B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
((((b4 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b5 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b6 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b7 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b8 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' b9 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
meets EqClass (
z : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
b3 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F,
J being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) )
for
z,
u being ( ( ) ( )
Element of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) is
independent &
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
EqClass (
z : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
((((C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
(((b5 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b6 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b7 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b8 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' b9 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
= EqClass (
u : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
((((C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
(((b5 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b6 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b7 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b8 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' b9 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) holds
EqClass (
u : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
(CompF (A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,G : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
CompF (
b3 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
b2 : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
meets EqClass (
z : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
(CompF (B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,G : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
CompF (
b4 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
b2 : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F,
J,
M being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
CompF (
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
= (((((B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F,
J,
M being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
CompF (
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
= (((((A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F,
J,
M being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
CompF (
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
= (((((A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F,
J,
M being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
CompF (
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
= (((((A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F,
J,
M being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
CompF (
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
= (((((A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F,
J,
M being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
CompF (
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
= (((((A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F,
J,
M being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
CompF (
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
= (((((A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F,
J,
M being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
CompF (
M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
= (((((A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
A,
B,
C,
D,
E,
F,
J,
M being ( ( ) ( )
set )
for
h being ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
for
A9,
B9,
C9,
D9,
E9,
F9,
J9,
M9 being ( ( ) ( )
set ) st
A : ( ( ) ( )
set )
<> B : ( ( ) ( )
set ) &
A : ( ( ) ( )
set )
<> C : ( ( ) ( )
set ) &
A : ( ( ) ( )
set )
<> D : ( ( ) ( )
set ) &
A : ( ( ) ( )
set )
<> E : ( ( ) ( )
set ) &
A : ( ( ) ( )
set )
<> F : ( ( ) ( )
set ) &
A : ( ( ) ( )
set )
<> J : ( ( ) ( )
set ) &
B : ( ( ) ( )
set )
<> C : ( ( ) ( )
set ) &
B : ( ( ) ( )
set )
<> D : ( ( ) ( )
set ) &
B : ( ( ) ( )
set )
<> E : ( ( ) ( )
set ) &
B : ( ( ) ( )
set )
<> F : ( ( ) ( )
set ) &
B : ( ( ) ( )
set )
<> J : ( ( ) ( )
set ) &
B : ( ( ) ( )
set )
<> M : ( ( ) ( )
set ) &
C : ( ( ) ( )
set )
<> D : ( ( ) ( )
set ) &
C : ( ( ) ( )
set )
<> E : ( ( ) ( )
set ) &
C : ( ( ) ( )
set )
<> F : ( ( ) ( )
set ) &
C : ( ( ) ( )
set )
<> J : ( ( ) ( )
set ) &
C : ( ( ) ( )
set )
<> M : ( ( ) ( )
set ) &
D : ( ( ) ( )
set )
<> E : ( ( ) ( )
set ) &
D : ( ( ) ( )
set )
<> F : ( ( ) ( )
set ) &
D : ( ( ) ( )
set )
<> J : ( ( ) ( )
set ) &
D : ( ( ) ( )
set )
<> M : ( ( ) ( )
set ) &
E : ( ( ) ( )
set )
<> F : ( ( ) ( )
set ) &
E : ( ( ) ( )
set )
<> J : ( ( ) ( )
set ) &
E : ( ( ) ( )
set )
<> M : ( ( ) ( )
set ) &
F : ( ( ) ( )
set )
<> J : ( ( ) ( )
set ) &
F : ( ( ) ( )
set )
<> M : ( ( ) ( )
set ) &
J : ( ( ) ( )
set )
<> M : ( ( ) ( )
set ) &
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
= (((((((B : ( ( ) ( ) set ) .--> B9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b2 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) +* (C : ( ( ) ( ) set ) .--> C9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b3 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (D : ( ( ) ( ) set ) .--> D9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b4 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (E : ( ( ) ( ) set ) .--> E9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b5 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (F : ( ( ) ( ) set ) .--> F9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b6 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (J : ( ( ) ( ) set ) .--> J9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b7 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (M : ( ( ) ( ) set ) .--> M9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b8 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( (
Relation-like Function-like ) (
Relation-like Function-like )
set )
+* (A : ( ( ) ( ) set ) .--> A9 : ( ( ) ( ) set ) ) : ( ( ) (
trivial Relation-like {b1 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined Function-like one-to-one )
set ) : ( (
Relation-like Function-like ) (
Relation-like Function-like )
set ) holds
(
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. B : ( ( ) ( )
set ) : ( ( ) ( )
set )
= B9 : ( ( ) ( )
set ) &
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. C : ( ( ) ( )
set ) : ( ( ) ( )
set )
= C9 : ( ( ) ( )
set ) &
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. D : ( ( ) ( )
set ) : ( ( ) ( )
set )
= D9 : ( ( ) ( )
set ) &
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. E : ( ( ) ( )
set ) : ( ( ) ( )
set )
= E9 : ( ( ) ( )
set ) &
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. F : ( ( ) ( )
set ) : ( ( ) ( )
set )
= F9 : ( ( ) ( )
set ) &
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. J : ( ( ) ( )
set ) : ( ( ) ( )
set )
= J9 : ( ( ) ( )
set ) ) ;
theorem
for
A,
B,
C,
D,
E,
F,
J,
M being ( ( ) ( )
set )
for
h being ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
for
A9,
B9,
C9,
D9,
E9,
F9,
J9,
M9 being ( ( ) ( )
set ) st
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
= (((((((B : ( ( ) ( ) set ) .--> B9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b2 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) +* (C : ( ( ) ( ) set ) .--> C9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b3 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (D : ( ( ) ( ) set ) .--> D9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b4 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (E : ( ( ) ( ) set ) .--> E9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b5 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (F : ( ( ) ( ) set ) .--> F9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b6 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (J : ( ( ) ( ) set ) .--> J9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b7 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (M : ( ( ) ( ) set ) .--> M9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b8 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( (
Relation-like Function-like ) (
Relation-like Function-like )
set )
+* (A : ( ( ) ( ) set ) .--> A9 : ( ( ) ( ) set ) ) : ( ( ) (
trivial Relation-like {b1 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined Function-like one-to-one )
set ) : ( (
Relation-like Function-like ) (
Relation-like Function-like )
set ) holds
dom h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function) : ( ( ) ( )
set )
= {A : ( ( ) ( ) set ) ,B : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ,D : ( ( ) ( ) set ) ,E : ( ( ) ( ) set ) ,F : ( ( ) ( ) set ) ,J : ( ( ) ( ) set ) ,M : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) ;
theorem
for
A,
B,
C,
D,
E,
F,
J,
M being ( ( ) ( )
set )
for
h being ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
for
A9,
B9,
C9,
D9,
E9,
F9,
J9,
M9 being ( ( ) ( )
set ) st
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
= (((((((B : ( ( ) ( ) set ) .--> B9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b2 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) +* (C : ( ( ) ( ) set ) .--> C9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b3 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (D : ( ( ) ( ) set ) .--> D9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b4 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (E : ( ( ) ( ) set ) .--> E9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b5 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (F : ( ( ) ( ) set ) .--> F9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b6 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (J : ( ( ) ( ) set ) .--> J9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b7 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (M : ( ( ) ( ) set ) .--> M9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b8 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( (
Relation-like Function-like ) (
Relation-like Function-like )
set )
+* (A : ( ( ) ( ) set ) .--> A9 : ( ( ) ( ) set ) ) : ( ( ) (
trivial Relation-like {b1 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined Function-like one-to-one )
set ) : ( (
Relation-like Function-like ) (
Relation-like Function-like )
set ) holds
rng h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function) : ( ( ) ( )
set )
= {(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . A : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . B : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . C : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . D : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . E : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . F : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . J : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . M : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F,
J,
M being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) )
for
z,
u being ( ( ) ( )
Element of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) is
independent &
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
(EqClass (u : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ,((((((B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( ) ( )
Element of
(((((b4 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b5 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b6 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b7 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b8 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b9 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' b10 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
/\ (EqClass (z : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ,A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( ) ( )
Element of
b3 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
bool b1 : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( non
empty )
set ) )
<> {} : ( ( ) (
empty )
set ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F,
J,
M being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) )
for
z,
u being ( ( ) ( )
Element of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) is
independent &
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
EqClass (
z : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
(((((C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
((((b5 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b6 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b7 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b8 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b9 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' b10 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
= EqClass (
u : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
(((((C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
((((b5 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b6 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b7 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b8 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b9 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' b10 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) holds
EqClass (
u : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
(CompF (A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,G : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
CompF (
b3 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
b2 : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
meets EqClass (
z : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
(CompF (B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,G : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
CompF (
b4 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
b2 : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F,
J,
M,
N being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,N : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
CompF (
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
= ((((((B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F,
J,
M,
N being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,N : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
CompF (
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
= ((((((A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F,
J,
M,
N being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,N : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
CompF (
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
= ((((((A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F,
J,
M,
N being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,N : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
CompF (
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
= ((((((A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F,
J,
M,
N being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,N : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
CompF (
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
= ((((((A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F,
J,
M,
N being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,N : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
CompF (
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
= ((((((A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F,
J,
M,
N being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,N : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
CompF (
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
= ((((((A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F,
J,
M,
N being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,N : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
CompF (
M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
= ((((((A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F,
J,
M,
N being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,N : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
CompF (
N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
= ((((((A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
A,
B,
C,
D,
E,
F,
J,
M,
N being ( ( ) ( )
set )
for
h being ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
for
A9,
B9,
C9,
D9,
E9,
F9,
J9,
M9,
N9 being ( ( ) ( )
set ) st
A : ( ( ) ( )
set )
<> B : ( ( ) ( )
set ) &
A : ( ( ) ( )
set )
<> C : ( ( ) ( )
set ) &
A : ( ( ) ( )
set )
<> D : ( ( ) ( )
set ) &
A : ( ( ) ( )
set )
<> E : ( ( ) ( )
set ) &
A : ( ( ) ( )
set )
<> F : ( ( ) ( )
set ) &
A : ( ( ) ( )
set )
<> J : ( ( ) ( )
set ) &
A : ( ( ) ( )
set )
<> M : ( ( ) ( )
set ) &
A : ( ( ) ( )
set )
<> N : ( ( ) ( )
set ) &
B : ( ( ) ( )
set )
<> C : ( ( ) ( )
set ) &
B : ( ( ) ( )
set )
<> D : ( ( ) ( )
set ) &
B : ( ( ) ( )
set )
<> E : ( ( ) ( )
set ) &
B : ( ( ) ( )
set )
<> F : ( ( ) ( )
set ) &
B : ( ( ) ( )
set )
<> J : ( ( ) ( )
set ) &
B : ( ( ) ( )
set )
<> M : ( ( ) ( )
set ) &
B : ( ( ) ( )
set )
<> N : ( ( ) ( )
set ) &
C : ( ( ) ( )
set )
<> D : ( ( ) ( )
set ) &
C : ( ( ) ( )
set )
<> E : ( ( ) ( )
set ) &
C : ( ( ) ( )
set )
<> F : ( ( ) ( )
set ) &
C : ( ( ) ( )
set )
<> J : ( ( ) ( )
set ) &
C : ( ( ) ( )
set )
<> M : ( ( ) ( )
set ) &
C : ( ( ) ( )
set )
<> N : ( ( ) ( )
set ) &
D : ( ( ) ( )
set )
<> E : ( ( ) ( )
set ) &
D : ( ( ) ( )
set )
<> F : ( ( ) ( )
set ) &
D : ( ( ) ( )
set )
<> J : ( ( ) ( )
set ) &
D : ( ( ) ( )
set )
<> M : ( ( ) ( )
set ) &
D : ( ( ) ( )
set )
<> N : ( ( ) ( )
set ) &
E : ( ( ) ( )
set )
<> F : ( ( ) ( )
set ) &
E : ( ( ) ( )
set )
<> J : ( ( ) ( )
set ) &
E : ( ( ) ( )
set )
<> M : ( ( ) ( )
set ) &
E : ( ( ) ( )
set )
<> N : ( ( ) ( )
set ) &
F : ( ( ) ( )
set )
<> J : ( ( ) ( )
set ) &
F : ( ( ) ( )
set )
<> M : ( ( ) ( )
set ) &
F : ( ( ) ( )
set )
<> N : ( ( ) ( )
set ) &
J : ( ( ) ( )
set )
<> M : ( ( ) ( )
set ) &
J : ( ( ) ( )
set )
<> N : ( ( ) ( )
set ) &
M : ( ( ) ( )
set )
<> N : ( ( ) ( )
set ) &
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
= ((((((((B : ( ( ) ( ) set ) .--> B9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b2 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) +* (C : ( ( ) ( ) set ) .--> C9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b3 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (D : ( ( ) ( ) set ) .--> D9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b4 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (E : ( ( ) ( ) set ) .--> E9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b5 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (F : ( ( ) ( ) set ) .--> F9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b6 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (J : ( ( ) ( ) set ) .--> J9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b7 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (M : ( ( ) ( ) set ) .--> M9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b8 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (N : ( ( ) ( ) set ) .--> N9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b9 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( (
Relation-like Function-like ) (
Relation-like Function-like )
set )
+* (A : ( ( ) ( ) set ) .--> A9 : ( ( ) ( ) set ) ) : ( ( ) (
trivial Relation-like {b1 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined Function-like one-to-one )
set ) : ( (
Relation-like Function-like ) (
Relation-like Function-like )
set ) holds
(
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. A : ( ( ) ( )
set ) : ( ( ) ( )
set )
= A9 : ( ( ) ( )
set ) &
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. B : ( ( ) ( )
set ) : ( ( ) ( )
set )
= B9 : ( ( ) ( )
set ) &
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. C : ( ( ) ( )
set ) : ( ( ) ( )
set )
= C9 : ( ( ) ( )
set ) &
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. D : ( ( ) ( )
set ) : ( ( ) ( )
set )
= D9 : ( ( ) ( )
set ) &
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. E : ( ( ) ( )
set ) : ( ( ) ( )
set )
= E9 : ( ( ) ( )
set ) &
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. F : ( ( ) ( )
set ) : ( ( ) ( )
set )
= F9 : ( ( ) ( )
set ) &
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. J : ( ( ) ( )
set ) : ( ( ) ( )
set )
= J9 : ( ( ) ( )
set ) &
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. M : ( ( ) ( )
set ) : ( ( ) ( )
set )
= M9 : ( ( ) ( )
set ) &
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. N : ( ( ) ( )
set ) : ( ( ) ( )
set )
= N9 : ( ( ) ( )
set ) ) ;
theorem
for
A,
B,
C,
D,
E,
F,
J,
M,
N being ( ( ) ( )
set )
for
h being ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
for
A9,
B9,
C9,
D9,
E9,
F9,
J9,
M9,
N9 being ( ( ) ( )
set ) st
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
= ((((((((B : ( ( ) ( ) set ) .--> B9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b2 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) +* (C : ( ( ) ( ) set ) .--> C9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b3 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (D : ( ( ) ( ) set ) .--> D9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b4 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (E : ( ( ) ( ) set ) .--> E9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b5 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (F : ( ( ) ( ) set ) .--> F9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b6 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (J : ( ( ) ( ) set ) .--> J9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b7 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (M : ( ( ) ( ) set ) .--> M9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b8 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (N : ( ( ) ( ) set ) .--> N9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b9 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( (
Relation-like Function-like ) (
Relation-like Function-like )
set )
+* (A : ( ( ) ( ) set ) .--> A9 : ( ( ) ( ) set ) ) : ( ( ) (
trivial Relation-like {b1 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined Function-like one-to-one )
set ) : ( (
Relation-like Function-like ) (
Relation-like Function-like )
set ) holds
dom h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function) : ( ( ) ( )
set )
= {A : ( ( ) ( ) set ) ,B : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ,D : ( ( ) ( ) set ) ,E : ( ( ) ( ) set ) ,F : ( ( ) ( ) set ) ,J : ( ( ) ( ) set ) ,M : ( ( ) ( ) set ) ,N : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) ;
theorem
for
A,
B,
C,
D,
E,
F,
J,
M,
N being ( ( ) ( )
set )
for
h being ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
for
A9,
B9,
C9,
D9,
E9,
F9,
J9,
M9,
N9 being ( ( ) ( )
set ) st
h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
= ((((((((B : ( ( ) ( ) set ) .--> B9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b2 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) +* (C : ( ( ) ( ) set ) .--> C9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b3 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (D : ( ( ) ( ) set ) .--> D9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b4 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (E : ( ( ) ( ) set ) .--> E9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b5 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (F : ( ( ) ( ) set ) .--> F9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b6 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (J : ( ( ) ( ) set ) .--> J9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b7 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (M : ( ( ) ( ) set ) .--> M9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b8 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) +* (N : ( ( ) ( ) set ) .--> N9 : ( ( ) ( ) set ) ) : ( ( ) ( trivial Relation-like {b9 : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) -defined Function-like one-to-one ) set ) ) : ( (
Relation-like Function-like ) (
Relation-like Function-like )
set )
+* (A : ( ( ) ( ) set ) .--> A9 : ( ( ) ( ) set ) ) : ( ( ) (
trivial Relation-like {b1 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined Function-like one-to-one )
set ) : ( (
Relation-like Function-like ) (
Relation-like Function-like )
set ) holds
rng h : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function) : ( ( ) ( )
set )
= {(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . A : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . B : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . C : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . D : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . E : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . F : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . J : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . M : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,(h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . N : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F,
J,
M,
N being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) )
for
z,
u being ( ( ) ( )
Element of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) is
independent &
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,N : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) holds
(EqClass (u : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ,(((((((B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' N : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( ) ( )
Element of
((((((b4 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b5 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b6 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b7 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b8 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b9 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b10 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' b11 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
/\ (EqClass (z : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ,A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( ) ( )
Element of
b3 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
bool b1 : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( non
empty )
set ) )
<> {} : ( ( ) (
empty )
set ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
G being ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
for
A,
B,
C,
D,
E,
F,
J,
M,
N being ( ( ) ( non
empty with_non-empty_elements )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) )
for
z,
u being ( ( ) ( )
Element of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) is
independent &
G : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) )
= {A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,N : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
A : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
B : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
C : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
D : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
E : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
F : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
J : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
M : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
<> N : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) &
EqClass (
z : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
((((((C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' N : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
(((((b5 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b6 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b7 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b8 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b9 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b10 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' b11 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
= EqClass (
u : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
((((((C : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' D : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' E : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' F : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' J : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' M : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' N : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
(((((b5 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b6 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b7 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b8 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b9 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) '/\' b10 : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) )
'/\' b11 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) holds
EqClass (
u : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
(CompF (A : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,G : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
CompF (
b3 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
b2 : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
meets EqClass (
z : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
(CompF (B : ( ( ) ( non empty with_non-empty_elements ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,G : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
CompF (
b4 : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
b2 : ( ( ) ( )
Subset of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( non
empty with_non-empty_elements )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ;