begin
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b being ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'&' (('not' a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Element of K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) 'imp' b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= b : ( (
Function-like quasi_total ) (
Function-like 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 ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'&' (a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' ('not' b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Element of K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= 'not' a : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
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 ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
a : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' (b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' c : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= (a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'or' (a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
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 ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
a : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' (b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' c : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= (a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'&' (a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
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 ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' c : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= (a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'&' (b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
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 ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' c : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= (a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'or' (b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
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 ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' c : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= a : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' (b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
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 ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' c : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= a : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' (('not' b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Element of K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) 'or' c : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
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 ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
a : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' (b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' c : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= (a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' ('not' b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Element of K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' c : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
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 ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
a : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'&' (a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= a : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'&' b : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
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 ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'&' ('not' b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= ('not' a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'&' ('not' b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
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 ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'&' (b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= ((a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Element of K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) '&' (b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Element of K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'&' (a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
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 ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
a : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' b : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'<' (c : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' (c : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
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 ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
a : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'eqv' b : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'<' (a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'eqv' c : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'eqv' (b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'eqv' c : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
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 ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
a : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'eqv' b : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'<' (a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'eqv' (b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
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 ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
a : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'eqv' b : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'<' (c : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'eqv' (c : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
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 ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
a : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'eqv' b : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'<' (a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' c : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'eqv' (b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' c : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
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 ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
a : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'eqv' b : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'<' (a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' c : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'eqv' (b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' c : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
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 ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
a : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'<' ((a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'eqv' b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Element of K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) 'eqv' (b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'eqv' a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Element of K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'eqv' a : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
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 ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
a : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'<' ((a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Element of K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) 'eqv' (b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Element of K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'eqv' a : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;