begin
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 ) ) st
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 ) )
'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
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
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
K19(
K20(
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 ) ) ;
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 ) ) st
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
K19(
K20(
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 ) ) 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 ) )
'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
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b being ( (
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
K19(
K20(
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
K19(
K20(
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' 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
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b being ( (
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 ) )
'xor' 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
K19(
K20(
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 K19(K20(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 ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
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 ) ) '&' ('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 K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b being ( (
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
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) iff (
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
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
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' 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
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b being ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) st
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
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) holds
('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
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'eqv' ('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
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b,
c,
d being ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) st
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
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
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 ) )
'eqv' d : ( (
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
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
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 ) ) '&' 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
K19(
K20(
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 ) ) '&' d : ( ( 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
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b,
c,
d being ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) st
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
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
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 ) )
'eqv' d : ( (
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
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
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' 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
K19(
K20(
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' d : ( ( 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
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b,
c,
d being ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) st
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
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
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 ) )
'eqv' d : ( (
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
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
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' 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
K19(
K20(
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' d : ( ( 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
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b,
c,
d being ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) st
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
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
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 ) )
'eqv' d : ( (
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
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
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' 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
K19(
K20(
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' d : ( ( 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
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
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 ) )
for
G being ( ( ) ( )
Subset of ( ( ) ( )
set ) )
for
PA being ( ( ) ( )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) holds
All (
(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
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ,
PA : ( ( ) ( )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= (All ((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 K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,PA : ( ( ) ( ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,G : ( ( ) ( ) Subset of ( ( ) ( ) set ) ) )) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'&' (All ((b : ( ( 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 K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,PA : ( ( ) ( ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,G : ( ( ) ( ) Subset of ( ( ) ( ) set ) ) )) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a being ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
for
G being ( ( ) ( )
Subset of ( ( ) ( )
set ) )
for
PA,
PB being ( ( ) ( )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) holds
All (
a : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) ,
PA : ( ( ) ( )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'<' Ex (
a : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) ,
PB : ( ( ) ( )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
u being ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
for
G being ( ( ) ( )
Subset of ( ( ) ( )
set ) )
for
PA being ( ( ) ( )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
a : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
'imp' u : ( (
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
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) holds
(All (a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ,PA : ( ( ) ( ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,G : ( ( ) ( ) Subset of ( ( ) ( ) set ) ) )) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' u : ( (
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
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
u being ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
for
G being ( ( ) ( )
Subset of ( ( ) ( )
set ) )
for
PA,
PB being ( ( ) ( )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
u : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
is_independent_of PB : ( ( ) ( )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) holds
All (
u : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) ,
PA : ( ( ) ( )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'<' All (
u : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) ,
PB : ( ( ) ( )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
u being ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
for
G being ( ( ) ( )
Subset of ( ( ) ( )
set ) )
for
PA,
PB being ( ( ) ( )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
u : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
is_independent_of PA : ( ( ) ( )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) holds
Ex (
u : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) ,
PA : ( ( ) ( )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'<' Ex (
u : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) ,
PB : ( ( ) ( )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b being ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
for
G being ( ( ) ( )
Subset of ( ( ) ( )
set ) )
for
PA being ( ( ) ( )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) holds
All (
(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
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ,
PA : ( ( ) ( )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'<' (All (a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ,PA : ( ( ) ( ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,G : ( ( ) ( ) Subset of ( ( ) ( ) set ) ) )) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'eqv' (All (b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ,PA : ( ( ) ( ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,G : ( ( ) ( ) Subset of ( ( ) ( ) set ) ) )) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b being ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
for
G being ( ( ) ( )
Subset of ( ( ) ( )
set ) )
for
PA being ( ( ) ( )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) holds
All (
(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
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ,
PA : ( ( ) ( )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
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 ) )
'&' (All (b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ,PA : ( ( ) ( ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,G : ( ( ) ( ) Subset of ( ( ) ( ) set ) ) )) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
u being ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
for
G being ( ( ) ( )
Subset of ( ( ) ( )
set ) )
for
PA being ( ( ) ( )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) holds
(All (a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ,PA : ( ( ) ( ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,G : ( ( ) ( ) Subset of ( ( ) ( ) set ) ) )) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'imp' u : ( (
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
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'<' Ex (
(a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) 'imp' u : ( ( 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
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ,
PA : ( ( ) ( )
a_partition of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
G : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b being ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
for
G being ( ( ) ( )
Subset of ( ( ) ( )
set ) )
for
PA being ( ( ) ( )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
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
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) holds
(All (a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ,PA : ( ( ) ( ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,G : ( ( ) ( ) Subset of ( ( ) ( ) set ) ) )) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'eqv' (All (b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ,PA : ( ( ) ( ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,G : ( ( ) ( ) Subset of ( ( ) ( ) set ) ) )) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Y being ( ( non
empty ) ( non
empty )
set )
for
a,
b being ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Function of
Y : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) )
for
G being ( ( ) ( )
Subset of ( ( ) ( )
set ) )
for
PA being ( ( ) ( )
a_partition of
Y : ( ( non
empty ) ( non
empty )
set ) ) st
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
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) holds
(Ex (a : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ,PA : ( ( ) ( ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,G : ( ( ) ( ) Subset of ( ( ) ( ) set ) ) )) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
'eqv' (Ex (b : ( ( Function-like quasi_total ) ( Function-like quasi_total boolean-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ,PA : ( ( ) ( ) a_partition of b1 : ( ( non empty ) ( non empty ) set ) ) ,G : ( ( ) ( ) Subset of ( ( ) ( ) set ) ) )) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= I_el Y : ( ( non
empty ) ( non
empty )
set ) : ( (
Function-like quasi_total ) (
Function-like quasi_total boolean-valued )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
BOOLEAN : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;