begin
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b being ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
'not' (a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
= a : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'&' ('not' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
b,
a being ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(('not' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) 'imp' ('not' a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
'imp' (a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
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 ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
a : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' b : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
= ('not' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
'imp' ('not' a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
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 ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
a : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'eqv' b : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
= ('not' a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
'eqv' ('not' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
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 ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
a : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' b : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
= a : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' (a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
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 ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
a : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'eqv' b : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
= (a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
'imp' (a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
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 ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
a : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' (b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
= b : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' (a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
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 ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
a : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' (b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
= (a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
'imp' (a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
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 ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
a : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'&' (b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'xor' c : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
= (a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
'xor' (a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' c : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
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 ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
'imp' (b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
= b : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' a : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
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 ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
'&' (('not' a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) 'or' ('not' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
= (('not' a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) '&' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
'or' (a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' ('not' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
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 ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
'or' (('not' a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) '&' ('not' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
= (('not' a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) 'or' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
'&' (a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' ('not' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
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 ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
a : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'xor' (b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'xor' c : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
= (a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'xor' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
'xor' c : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
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 ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
a : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'eqv' (b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'eqv' c : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
= (a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'eqv' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
'eqv' c : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
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 ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
((a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) '&' a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
'imp' b : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
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 ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
'or' (c : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
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 ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
'or' (('not' a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) 'imp' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
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 ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
'or' (a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' ('not' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
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 ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
('not' a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
'imp' (('not' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) 'eqv' (b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
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 ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
'imp' (((a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) 'imp' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) 'imp' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
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 ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
a : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' b : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
= a : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'eqv' (a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
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 ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
( (
a : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' b : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) &
b : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' a : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) ) iff
a : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
= b : ( (
V12()
quasi_total ) (
V12()
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 ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
a : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' ((a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) 'imp' a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
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 ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
a : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
= (b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
'&' (('not' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) 'imp' a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
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 ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
a : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'&' b : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
= 'not' (a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' ('not' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
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 ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
a : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'or' b : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
= (a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
'imp' b : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
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 ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
'imp' (a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
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,
d being ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' (b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
'imp' ((d : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) 'imp' (a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' (d : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
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 ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(((a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) '&' a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) '&' c : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
'imp' b : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
b,
c,
a being ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
'imp' ((a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) 'imp' c : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
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 ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
((a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) 'imp' c : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
'imp' ((a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) 'imp' (c : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
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 ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
'imp' ((a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' c : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) 'imp' (b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' c : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
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 ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
((a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) '&' (a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' c : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
'imp' (b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' c : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
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 ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
(a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) '&' (a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
'&' (b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
'<' c : ( (
V12()
quasi_total ) (
V12()
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 being ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) holds
((a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) '&' (a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) M2(K10(K11(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) )) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
'&' (b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' c : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
'<' ('not' a : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ))
'imp' (b : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'or' c : ( ( V12() quasi_total ) ( V12() quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) : ( (
V12()
quasi_total ) (
V12()
quasi_total boolean-valued )
M2(
K10(
K11(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )) ;