:: SETLIM_1 semantic presentation

begin

theorem :: SETLIM_1:1
for n, m being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) )
for Y being ( ( ) ( ) set )
for f being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b3 : ( ( ) ( ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b3 : ( ( ) ( ) set ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b3 : ( ( ) ( ) set ) ) ) Function of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,Y : ( ( ) ( ) set ) ) holds f : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b3 : ( ( ) ( ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b3 : ( ( ) ( ) set ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b3 : ( ( ) ( ) set ) ) ) Function of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b3 : ( ( ) ( ) set ) ) . (n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) + m : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of b3 : ( ( ) ( ) set ) ) in { (f : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b3 : ( ( ) ( ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b3 : ( ( ) ( ) set ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b3 : ( ( ) ( ) set ) ) ) Function of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b3 : ( ( ) ( ) set ) ) . k : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of b3 : ( ( ) ( ) set ) ) where k is ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) <= k : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) } ;

theorem :: SETLIM_1:2
for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) )
for Y being ( ( ) ( ) set )
for f being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( ) ( ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( ) ( ) set ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( ) ( ) set ) ) ) Function of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,Y : ( ( ) ( ) set ) ) holds { (f : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( ) ( ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( ) ( ) set ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( ) ( ) set ) ) ) Function of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( ) ( ) set ) ) . k1 : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( ) ( ) set ) ) where k1 is ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) <= k1 : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) } = { (f : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( ) ( ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( ) ( ) set ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( ) ( ) set ) ) ) Function of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( ) ( ) set ) ) . k2 : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( ) ( ) set ) ) where k2 is ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) + 1 : ( ( ) ( non empty V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) <= k2 : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) } \/ {(f : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( ) ( ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( ) ( ) set ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( ) ( ) set ) ) ) Function of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( ) ( ) set ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( ) ( ) set ) ) } : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ;

theorem :: SETLIM_1:3
for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) )
for Y, x being ( ( ) ( ) set )
for f being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( ) ( ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( ) ( ) set ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( ) ( ) set ) ) ) Function of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,Y : ( ( ) ( ) set ) ) holds
( ( for k1 being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds x : ( ( ) ( ) set ) in f : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( ) ( ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( ) ( ) set ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( ) ( ) set ) ) ) Function of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( ) ( ) set ) ) . (n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) + k1 : ( ( ) ( ) set ) ) : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( ) ( ) set ) ) ) iff for Z being ( ( ) ( ) set ) st Z : ( ( ) ( ) set ) in { (f : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( ) ( ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( ) ( ) set ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( ) ( ) set ) ) ) Function of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( ) ( ) set ) ) . k2 : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( ) ( ) set ) ) where k2 is ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) <= k2 : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) } holds
x : ( ( ) ( ) set ) in Z : ( ( ) ( ) set ) ) ;

theorem :: SETLIM_1:4
for x being ( ( ) ( ) set )
for Y being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,Y : ( ( non empty ) ( non empty ) set ) ) holds
( x : ( ( ) ( ) set ) in rng f : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V59() ) set ) ) iff ex n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st x : ( ( ) ( ) set ) = f : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: SETLIM_1:5
for Y being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) Function of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,Y : ( ( non empty ) ( non empty ) set ) ) holds rng f : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) Function of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V59() ) set ) ) = { (f : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) Function of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . k : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) where k is ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : 0 : ( ( ) ( empty V7() V8() V9() V11() V12() V13() V14() ext-real Relation-like non-empty empty-yielding V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) <= k : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) } ;

theorem :: SETLIM_1:6
for k being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) )
for Y being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,Y : ( ( non empty ) ( non empty ) set ) ) holds rng (f : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ^\ k : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Element of bool [:NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty V59() ) set ) ) : ( ( ) ( non empty ) Element of bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V59() ) set ) ) = { (f : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) where n is ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : k : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) <= n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) } ;

theorem :: SETLIM_1:7
for X, x being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds
( x : ( ( ) ( ) set ) in meet (rng B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) : ( ( ) ( non empty V59() ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) iff for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds x : ( ( ) ( ) set ) in B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ;

theorem :: SETLIM_1:8
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds Intersection B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) = meet (rng B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) : ( ( ) ( non empty V59() ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:9
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds Intersection B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) c= Union B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:10
for X being ( ( ) ( ) set )
for A being ( ( ) ( ) Subset of ( ( ) ( non empty V59() ) set ) )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st ( for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) = A : ( ( ) ( ) Subset of ( ( ) ( non empty V59() ) set ) ) ) holds
Union B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) = A : ( ( ) ( ) Subset of ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:11
for X being ( ( ) ( ) set )
for A being ( ( ) ( ) Subset of ( ( ) ( non empty V59() ) set ) )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st ( for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) = A : ( ( ) ( ) Subset of ( ( ) ( non empty V59() ) set ) ) ) holds
Intersection B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) = A : ( ( ) ( ) Subset of ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:12
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is constant holds
Union B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) = Intersection B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:13
for X being ( ( ) ( ) set )
for A being ( ( ) ( ) Subset of ( ( ) ( non empty V59() ) set ) )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is constant & the_value_of B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) set ) = A : ( ( ) ( ) Subset of ( ( ) ( non empty V59() ) set ) ) holds
for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds union { (B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . k : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) where k is ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) <= k : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) } : ( ( ) ( ) set ) = A : ( ( ) ( ) Subset of ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:14
for X being ( ( ) ( ) set )
for A being ( ( ) ( ) Subset of ( ( ) ( non empty V59() ) set ) )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is constant & the_value_of B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) set ) = A : ( ( ) ( ) Subset of ( ( ) ( non empty V59() ) set ) ) holds
for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds meet { (B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . k : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) where k is ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) <= k : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) } : ( ( ) ( ) set ) = A : ( ( ) ( ) Subset of ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:15
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) )
for f being ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) st dom f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) : ( ( ) ( ) set ) = NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) & ( for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) set ) = meet { (B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . k : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) where k is ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) <= k : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) } : ( ( ) ( ) set ) ) holds
f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) is ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ;

theorem :: SETLIM_1:16
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) )
for f being ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) st dom f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) : ( ( ) ( ) set ) = NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) & ( for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) set ) = union { (B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . k : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) where k is ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) <= k : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) } : ( ( ) ( ) set ) ) holds
f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) is ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) Function of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ;

definition
let X be ( ( ) ( ) set ) ;
let B be ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ;
attr B is monotone means :: SETLIM_1:def 1
( not B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) Element of bool [:NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,(bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty V59() ) set ) ) is V50() or not B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) Element of bool [:NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,(bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty V59() ) set ) ) is V49() );
end;

definition
let B be ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) ;
func inferior_setsequence B -> ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) means :: SETLIM_1:def 2
( dom it : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool B : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool B : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool B : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool B : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool B : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool B : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) Element of bool [:NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,(bool B : ( ( ) ( ) set ) ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool B : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty V59() ) set ) ) : ( ( ) ( non empty ) set ) = NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) & ( for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds it : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool B : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool B : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool B : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool B : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool B : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool B : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) Element of bool [:NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,(bool B : ( ( ) ( ) set ) ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool B : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty V59() ) set ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) set ) = meet { (B : ( ( ) ( ) set ) . k : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) set ) where k is ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) <= k : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) } : ( ( ) ( ) set ) ) );
end;

definition
let X be ( ( ) ( ) set ) ;
let B be ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ;
:: original: inferior_setsequence
redefine func inferior_setsequence B -> ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ;
end;

definition
let B be ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) ;
func superior_setsequence B -> ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) means :: SETLIM_1:def 3
( dom it : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool B : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool B : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool B : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool B : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool B : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool B : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) Element of bool [:NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,(bool B : ( ( ) ( ) set ) ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool B : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty V59() ) set ) ) : ( ( ) ( non empty ) set ) = NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) & ( for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds it : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool B : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool B : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool B : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool B : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool B : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool B : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) Element of bool [:NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,(bool B : ( ( ) ( ) set ) ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool B : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty V59() ) set ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) set ) = union { (B : ( ( ) ( ) set ) . k : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) set ) where k is ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) <= k : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) } : ( ( ) ( ) set ) ) );
end;

definition
let X be ( ( ) ( ) set ) ;
let B be ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ;
:: original: superior_setsequence
redefine func superior_setsequence B -> ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ;
end;

theorem :: SETLIM_1:17
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds (inferior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . 0 : ( ( ) ( empty V7() V8() V9() V11() V12() V13() V14() ext-real Relation-like non-empty empty-yielding V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) = Intersection B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:18
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds (superior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . 0 : ( ( ) ( empty V7() V8() V9() V11() V12() V13() V14() ext-real Relation-like non-empty empty-yielding V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) = Union B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:19
for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) )
for X, x being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds
( x : ( ( ) ( ) set ) in (inferior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) iff for k being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds x : ( ( ) ( ) set ) in B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . (n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) + k : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ;

theorem :: SETLIM_1:20
for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) )
for X, x being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds
( x : ( ( ) ( ) set ) in (superior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) iff ex k being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st x : ( ( ) ( ) set ) in B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . (n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) + k : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ;

theorem :: SETLIM_1:21
for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) )
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds (inferior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) = ((inferior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . (n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) + 1 : ( ( ) ( non empty V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) /\ (B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) M7(b2 : ( ( ) ( ) set ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) )) ;

theorem :: SETLIM_1:22
for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) )
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds (superior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) = ((superior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . (n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) + 1 : ( ( ) ( non empty V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) \/ (B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) M7(b2 : ( ( ) ( ) set ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) )) ;

theorem :: SETLIM_1:23
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds inferior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is V50() ;

theorem :: SETLIM_1:24
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds superior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is V49() ;

theorem :: SETLIM_1:25
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds
( inferior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is monotone & superior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is monotone ) ;

registration
let X be ( ( ) ( ) set ) ;
let A be ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ;
cluster inferior_setsequence A : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) Element of bool [:NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,(bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty V59() ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) -> Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V50() for ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ;
end;

registration
let X be ( ( ) ( ) set ) ;
let A be ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ;
cluster superior_setsequence A : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) Element of bool [:NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,(bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty V59() ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) -> Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V49() for ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ;
end;

theorem :: SETLIM_1:26
for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) )
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds Intersection B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) c= (inferior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V50() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ;

theorem :: SETLIM_1:27
for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) )
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds (superior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V49() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) c= Union B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:28
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) )
for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds { (B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . k : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) where k is ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) <= k : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) } is ( ( ) ( ) Subset-Family of ) ;

theorem :: SETLIM_1:29
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds Union B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) = (Intersection (Complement B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) Element of bool [:NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,(bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ` : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:30
for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) )
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds (inferior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V50() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) = ((superior_setsequence (Complement B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) Element of bool [:NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,(bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V49() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ` : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:31
for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) )
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds (superior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V49() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) = ((inferior_setsequence (Complement B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) Element of bool [:NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,(bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V50() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ` : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:32
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds Complement (inferior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V50() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) Element of bool [:NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,(bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty V59() ) set ) ) = superior_setsequence (Complement B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) Element of bool [:NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,(bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty V59() ) set ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V49() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ;

theorem :: SETLIM_1:33
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds Complement (superior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V49() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) Element of bool [:NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,(bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty V59() ) set ) ) = inferior_setsequence (Complement B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) Element of bool [:NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,(bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty V59() ) set ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V50() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ;

theorem :: SETLIM_1:34
for X being ( ( ) ( ) set )
for A3, A1, A2 being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st ( for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds A3 : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) = (A1 : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) \/ (A2 : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) M7(b1 : ( ( ) ( ) set ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) )) ) holds
for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds ((inferior_setsequence A1 : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V50() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) \/ ((inferior_setsequence A2 : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V50() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) M7(b1 : ( ( ) ( ) set ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) )) c= (inferior_setsequence A3 : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V50() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ;

theorem :: SETLIM_1:35
for X being ( ( ) ( ) set )
for A3, A1, A2 being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st ( for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds A3 : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) = (A1 : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) /\ (A2 : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) M7(b1 : ( ( ) ( ) set ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) )) ) holds
for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds (inferior_setsequence A3 : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V50() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) = ((inferior_setsequence A1 : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V50() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) /\ ((inferior_setsequence A2 : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V50() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) M7(b1 : ( ( ) ( ) set ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) )) ;

theorem :: SETLIM_1:36
for X being ( ( ) ( ) set )
for A3, A1, A2 being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st ( for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds A3 : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) = (A1 : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) \/ (A2 : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) M7(b1 : ( ( ) ( ) set ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) )) ) holds
for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds (superior_setsequence A3 : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V49() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) = ((superior_setsequence A1 : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V49() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) \/ ((superior_setsequence A2 : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V49() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) M7(b1 : ( ( ) ( ) set ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) )) ;

theorem :: SETLIM_1:37
for X being ( ( ) ( ) set )
for A3, A1, A2 being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st ( for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds A3 : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) = (A1 : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) /\ (A2 : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) M7(b1 : ( ( ) ( ) set ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) )) ) holds
for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds (superior_setsequence A3 : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V49() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) c= ((superior_setsequence A1 : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V49() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) /\ ((superior_setsequence A2 : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V49() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) M7(b1 : ( ( ) ( ) set ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) )) ;

theorem :: SETLIM_1:38
for X being ( ( ) ( ) set )
for A being ( ( ) ( ) Subset of ( ( ) ( non empty V59() ) set ) )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is constant & the_value_of B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) set ) = A : ( ( ) ( ) Subset of ( ( ) ( non empty V59() ) set ) ) holds
for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds (inferior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V50() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) = A : ( ( ) ( ) Subset of ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:39
for X being ( ( ) ( ) set )
for A being ( ( ) ( ) Subset of ( ( ) ( non empty V59() ) set ) )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is constant & the_value_of B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) set ) = A : ( ( ) ( ) Subset of ( ( ) ( non empty V59() ) set ) ) holds
for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds (superior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V49() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) = A : ( ( ) ( ) Subset of ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:40
for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) )
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is V50() holds
B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) c= (superior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V49() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . (n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) + 1 : ( ( ) ( non empty V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ;

theorem :: SETLIM_1:41
for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) )
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is V50() holds
(superior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V49() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) = (superior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V49() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . (n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) + 1 : ( ( ) ( non empty V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ;

theorem :: SETLIM_1:42
for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) )
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is V50() holds
(superior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V49() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) = Union B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:43
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is V50() holds
Intersection (superior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V49() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) = Union B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:44
for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) )
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is V50() holds
B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) c= (inferior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V50() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . (n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) + 1 : ( ( ) ( non empty V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ;

theorem :: SETLIM_1:45
for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) )
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is V50() holds
(inferior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V50() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) = B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ;

theorem :: SETLIM_1:46
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is V50() holds
inferior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V50() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) = B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ;

theorem :: SETLIM_1:47
for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) )
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is V49() holds
(superior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V49() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . (n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) + 1 : ( ( ) ( non empty V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) c= B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ;

theorem :: SETLIM_1:48
for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) )
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is V49() holds
(superior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V49() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) = B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ;

theorem :: SETLIM_1:49
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is V49() holds
superior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V49() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) = B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ;

theorem :: SETLIM_1:50
for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) )
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is V49() holds
(inferior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V50() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . (n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) + 1 : ( ( ) ( non empty V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) c= B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ;

theorem :: SETLIM_1:51
for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) )
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is V49() holds
(inferior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V50() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) = (inferior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V50() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . (n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) + 1 : ( ( ) ( non empty V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ;

theorem :: SETLIM_1:52
for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) )
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is V49() holds
(inferior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V50() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) = Intersection B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:53
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is V49() holds
Union (inferior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V50() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) = Intersection B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

definition
let X be ( ( ) ( ) set ) ;
let B be ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ;
redefine func lim_inf B equals :: SETLIM_1:def 4
 Union (inferior_setsequence B : ( ( ) ( ) set ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let B be ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ;
redefine func lim_sup B equals :: SETLIM_1:def 5
 Intersection (superior_setsequence B : ( ( ) ( ) set ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;
end;

notation
let X be ( ( ) ( ) set ) ;
let B be ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ;
synonym lim B for lim_sup B;
end;

theorem :: SETLIM_1:54
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds Intersection B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) c= lim_inf B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:55
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds lim_inf B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) = lim (inferior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V50() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:56
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds lim_sup B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) = lim (superior_setsequence B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V49() ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:57
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds lim_sup B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) = (lim_inf (Complement B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) Element of bool [:NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,(bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ` : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:58
for X being ( ( ) ( ) set )
for A being ( ( ) ( ) Subset of ( ( ) ( non empty V59() ) set ) )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is constant & the_value_of B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) set ) = A : ( ( ) ( ) Subset of ( ( ) ( non empty V59() ) set ) ) holds
( B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is convergent & lim B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) = A : ( ( ) ( ) Subset of ( ( ) ( non empty V59() ) set ) ) & lim_inf B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) = A : ( ( ) ( ) Subset of ( ( ) ( non empty V59() ) set ) ) & lim_sup B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) = A : ( ( ) ( ) Subset of ( ( ) ( non empty V59() ) set ) ) ) ;

theorem :: SETLIM_1:59
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is V50() holds
lim_sup B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) = Union B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:60
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is V50() holds
lim_inf B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) = Union B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:61
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is V49() holds
lim_sup B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) = Intersection B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:62
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is V49() holds
lim_inf B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) = Intersection B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:63
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is V50() holds
( B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is convergent & lim B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) = Union B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ;

theorem :: SETLIM_1:64
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is V49() holds
( B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is convergent & lim B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) = Intersection B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ;

theorem :: SETLIM_1:65
for X being ( ( ) ( ) set )
for B being ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is monotone holds
B : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) is convergent ;

definition
let X be ( ( ) ( ) set ) ;
let Si be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of X : ( ( ) ( ) set ) ) ;
let S be ( ( Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of X : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of X : ( ( ) ( ) set ) ) -valued bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of X : ( ( ) ( ) set ) ) ) ;
:: original: inferior_setsequence
redefine func inferior_setsequence S -> ( ( Si : ( ( ) ( ) set ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined Si : ( ( ) ( ) set ) -valued bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of Si : ( ( ) ( ) set ) ) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Si be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of X : ( ( ) ( ) set ) ) ;
let S be ( ( Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of X : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of X : ( ( ) ( ) set ) ) -valued bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of X : ( ( ) ( ) set ) ) ) ;
:: original: superior_setsequence
redefine func superior_setsequence S -> ( ( Si : ( ( ) ( ) set ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined Si : ( ( ) ( ) set ) -valued bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of Si : ( ( ) ( ) set ) ) ;
end;

theorem :: SETLIM_1:66
for X, x being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of X : ( ( ) ( ) set ) )
for S being ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) holds
( x : ( ( ) ( ) set ) in lim_sup S : ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) iff for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ex k being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st x : ( ( ) ( ) set ) in S : ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . (n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) + k : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ;

theorem :: SETLIM_1:67
for X, x being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of X : ( ( ) ( ) set ) )
for S being ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) holds
( x : ( ( ) ( ) set ) in lim_inf S : ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) iff ex n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) st
for k being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds x : ( ( ) ( ) set ) in S : ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . (n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) + k : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ;

theorem :: SETLIM_1:68
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of X : ( ( ) ( ) set ) )
for S being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) holds Intersection S : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) c= lim_inf S : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:69
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of X : ( ( ) ( ) set ) )
for S being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) holds lim_sup S : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) c= Union S : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:70
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of X : ( ( ) ( ) set ) )
for S being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) holds lim_inf S : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) c= lim_sup S : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:71
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of X : ( ( ) ( ) set ) )
for S being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) holds lim_inf S : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) = (lim_sup (Complement S : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) Element of bool [:NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,(bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ` : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:72
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of X : ( ( ) ( ) set ) )
for S being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) holds lim_sup S : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) = (lim_inf (Complement S : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) Element of bool [:NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ,(bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ` : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:73
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of X : ( ( ) ( ) set ) )
for S3, S1, S2 being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) st ( for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds S3 : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) = (S1 : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) \/ (S2 : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) M7(b1 : ( ( ) ( ) set ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) )) ) holds
(lim_inf S1 : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) \/ (lim_inf S2 : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) c= lim_inf S3 : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:74
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of X : ( ( ) ( ) set ) )
for S3, S1, S2 being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) st ( for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds S3 : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) = (S1 : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) /\ (S2 : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) M7(b1 : ( ( ) ( ) set ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) )) ) holds
lim_inf S3 : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) = (lim_inf S1 : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) /\ (lim_inf S2 : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:75
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of X : ( ( ) ( ) set ) )
for S3, S1, S2 being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) st ( for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds S3 : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) = (S1 : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) \/ (S2 : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) M7(b1 : ( ( ) ( ) set ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) )) ) holds
lim_sup S3 : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) = (lim_sup S1 : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) \/ (lim_sup S2 : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:76
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of X : ( ( ) ( ) set ) )
for S3, S1, S2 being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) st ( for n being ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) holds S3 : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) = (S1 : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) /\ (S2 : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( V7() V8() V9() V13() V14() ext-real V57() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) : ( ( ) ( ) M7(b1 : ( ( ) ( ) set ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) )) ) holds
lim_sup S3 : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) c= (lim_sup S1 : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) /\ (lim_sup S2 : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:77
for X being ( ( ) ( ) set )
for A being ( ( ) ( ) Subset of ( ( ) ( non empty V59() ) set ) )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of X : ( ( ) ( ) set ) )
for S being ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) st S : ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) is constant & the_value_of S : ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = A : ( ( ) ( ) Subset of ( ( ) ( non empty V59() ) set ) ) holds
( S : ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) is convergent & lim S : ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) = A : ( ( ) ( ) Subset of ( ( ) ( non empty V59() ) set ) ) & lim_inf S : ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) = A : ( ( ) ( ) Subset of ( ( ) ( non empty V59() ) set ) ) & lim_sup S : ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) = A : ( ( ) ( ) Subset of ( ( ) ( non empty V59() ) set ) ) ) ;

theorem :: SETLIM_1:78
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of X : ( ( ) ( ) set ) )
for S being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) st S : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) is V50() holds
lim_sup S : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) = Union S : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:79
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of X : ( ( ) ( ) set ) )
for S being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) st S : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) is V50() holds
lim_inf S : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) = Union S : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:80
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of X : ( ( ) ( ) set ) )
for S being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) st S : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) is V50() holds
( S : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) is convergent & lim S : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) = Union S : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ;

theorem :: SETLIM_1:81
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of X : ( ( ) ( ) set ) )
for S being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) st S : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) is V49() holds
lim_sup S : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) = Intersection S : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:82
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of X : ( ( ) ( ) set ) )
for S being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) st S : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) is V49() holds
lim_inf S : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) = Intersection S : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ;

theorem :: SETLIM_1:83
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of X : ( ( ) ( ) set ) )
for S being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) st S : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) is V49() holds
( S : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) is convergent & lim S : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) = Intersection S : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ;

theorem :: SETLIM_1:84
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of X : ( ( ) ( ) set ) )
for S being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) st S : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) is monotone holds
S : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) -valued Function-like V28( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) ) V32( NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( ) set ) : ( ( ) ( non empty V59() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V59() ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty V59() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V59() ) SigmaField of b1 : ( ( ) ( ) set ) ) ) is convergent ;