begin
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
a : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' (b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' (a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' ((b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) 'imp' (a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'eqv' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'eqv' (b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b,
c being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
((a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) 'imp' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' (a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' (b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b,
c being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' (b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' ((a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) 'imp' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b,
c being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' ((c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) 'imp' (c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' (a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b,
c being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
((a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) 'imp' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' ((a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) 'or' (b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b,
c being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' ((b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) 'imp' ((a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) 'imp' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b,
c being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
((a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) '&' (b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' ((a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) 'imp' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' (b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' ('not' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' ('not' a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b,
c being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
((a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) '&' (a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' (a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' (b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b,
c being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' (b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' ((a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) 'or' (a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b,
c being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
((a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) '&' (b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' ((a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) 'or' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b,
c being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
((a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) '&' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' ((a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) 'or' (b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) st
a : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'&' b : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) holds
a : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'or' b : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b,
c being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) st
a : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' b : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) holds
(a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' (b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b,
c being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) st
a : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' b : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) holds
(a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' (b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b,
c being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) st
c : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' a : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) &
c : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' b : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) holds
c : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' (a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b,
c being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) st
a : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' c : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) &
b : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' c : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) holds
(a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' c : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) st
a : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'or' b : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) &
'not' a : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) holds
b : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b,
c,
d being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) st
a : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' b : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) &
c : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' d : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) holds
(a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' (b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' d : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b,
c,
d being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) st
a : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' b : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) &
c : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' d : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) holds
(a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' (b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' d : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) st
(a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' ('not' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' ('not' a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) holds
a : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' b : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) st
a : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' ('not' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) holds
b : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' ('not' a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) st
('not' a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' b : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) holds
('not' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' a : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' (('not' a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) 'imp' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
('not' (a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' (('not' a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) '&' ('not' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(('not' a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) '&' ('not' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' ('not' (a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
('not' (a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' ('not' a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' (('not' a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) 'or' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' ('not' (a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' ('not' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
('not' (a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' ('not' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' (a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
('not' (a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' (('not' a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) 'or' ('not' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(('not' a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) 'or' ('not' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' ('not' (a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' (a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'eqv' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' (a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'eqv' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' (b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b,
c being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
((a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) 'or' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' (a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' (b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b,
c being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
((a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) '&' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' (a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' (b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b,
c being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' (b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' ((a : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' b : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) 'or' c : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;