begin
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 ) )
'&' (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 ) )
'<' 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 ) ) ;
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 ) )
'&' ('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 ) )
'<' '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 ) ) ;
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 ) )
'&' (('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 ) )
'<' b : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
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 ) )
'&' (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 ) )
'<' '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 ) ) ;
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 ) ) '&' 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 ) )
'<' 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 ) ) ;
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 ) )
'<' 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 ) ) ;
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 ) ) : ( (
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 ) ) '&' 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 ) ) : ( (
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 ) ) : ( (
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 ) ) '&' 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 ) ) ;
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 ) ) : ( (
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 ) )
'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 ) ) ;
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 ) ) : ( (
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 ) )
'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 ) ) ;
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 ) )
'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 ) )
'<' 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 ) ) ;
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 ) ) 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 ) )
'or' (c : ( ( 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 ) )
'<' 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 ) ) ;
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 ) )
'&' (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 ) )
'<' 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 ) ) ;
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 ) ) ) : ( (
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 ) ) '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 ) )
'<' 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 ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
c,
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' 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' ('not' 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 ) )
'<' ('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 ) ) ;
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 ) )
'&' (('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' 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 ) )
'<' 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 ) ) ;
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 ) ) 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 ) )
'&' (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 ) ) : ( (
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 ) ) '&' 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 ) ) ;
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 ) ) ) : ( (
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 ) ) '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 ) )
'<' 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 ) ) '&' 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 ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
c,
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' 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 ) )
'<' (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 ) ) ;
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 ) ) 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 ) )
'&' (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 ) ) : ( (
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 ) )
'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 ) ) ;
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 ) ) ) : ( (
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 ) ) '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 ) )
'<' 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 ) ) '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 ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a1,
b1,
c1,
a2,
b2,
c2 being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
((((b1 : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b2 : ( ( 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 ) ) '&' (c1 : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c2 : ( ( 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 ) ) '&' ((a1 : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' b1 : ( ( 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' c1 : ( ( 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 ) ) '&' ('not' (a2 : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' b2 : ( ( 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 ) )
'&' ('not' (a2 : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' c2 : ( ( 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 ) )
'<' a2 : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' a1 : ( (
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 ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a1,
b1,
c1,
a2,
b2,
c2 being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
((((((a1 : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' a2 : ( ( 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 ) ) '&' (b1 : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b2 : ( ( 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 ) ) '&' (c1 : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c2 : ( ( 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 ) ) '&' ((a1 : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' b1 : ( ( 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' c1 : ( ( 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 ) ) '&' ('not' (a2 : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' b2 : ( ( 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 ) ) '&' ('not' (a2 : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' c2 : ( ( 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 ) )
'&' ('not' (b2 : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' c2 : ( ( 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 ) )
'<' ((a2 : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' a1 : ( ( 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 ) ) '&' (b2 : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b1 : ( ( 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 ) )
'&' (c2 : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c1 : ( ( 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 ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a1,
b1,
a2,
b2 being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(((a1 : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' a2 : ( ( 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 ) ) '&' (b1 : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b2 : ( ( 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 ) ) '&' ('not' (a2 : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' b2 : ( ( 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' (a1 : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' b1 : ( ( 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
a1,
b1,
c1,
a2,
b2,
c2 being ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(((((a1 : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' a2 : ( ( 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 ) ) '&' (b1 : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b2 : ( ( 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 ) ) '&' (c1 : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c2 : ( ( 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 ) ) '&' ('not' (a2 : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' b2 : ( ( 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 ) ) '&' ('not' (a2 : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' c2 : ( ( 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 ) )
'&' ('not' (b2 : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' c2 : ( ( 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 ) )
'<' (('not' (a1 : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' b1 : ( ( 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 ) ) '&' ('not' (a1 : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' c1 : ( ( 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 ) )
'&' ('not' (b1 : ( ( V6() quasi_total ) ( V6() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' c1 : ( ( 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 ) ) ;
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 ) )
'<' a : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) &
(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 ) )
'<' b : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
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 ) ) 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 ) )
'&' 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 ) )
'<' a : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) &
((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 ) )
'&' 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 ) )
'<' b : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b,
c,
d,
e 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 ) ) '&' 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 ) )
'&' e : ( (
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 ) ) &
(((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 ) ) '&' 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 ) )
'&' e : ( (
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 ) ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b,
c,
d,
e,
f 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 ) ) '&' 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 ) ) '&' e : ( ( 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 ) )
'&' f : ( (
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 ) ) &
((((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 ) ) '&' 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 ) ) '&' e : ( ( 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 ) )
'&' f : ( (
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 ) ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b,
c,
d,
e,
f,
g 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 ) ) '&' 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 ) ) '&' e : ( ( 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 ) ) '&' f : ( ( 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 ) )
'&' g : ( (
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 ) ) &
(((((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 ) ) '&' 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 ) ) '&' e : ( ( 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 ) ) '&' f : ( ( 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 ) )
'&' g : ( (
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 ) ) ) ;
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 ) )
'<' 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 ) )
'<' d : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( 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 ) )
'&' 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 ) )
'&' 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 ) ) ;
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 ) )
'&' 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 ) ) holds
a : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'&' ('not' 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 ) )
'<' '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 ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
c,
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' 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 ) )
'&' (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 ) )
'<' c : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
c,
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' 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 ) )
'&' (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 ) )
'<' c : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
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 ) )
'<' 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 ) )
'<' d : ( (
V6()
quasi_total ) (
V6()
quasi_total boolean-valued )
Function of
b1 : ( ( 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' 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 ) ) ;