:: MEASURE8 semantic presentation

begin

theorem :: MEASURE8:1
for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) holds Ser seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) = Partial_Sums seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MEASURE8:2
for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) st seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) is nonnegative holds
( seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) is summable & SUM seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = Sum seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ) ;

theorem :: MEASURE8:3
for seq1, seq2, seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) st seq1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) is nonnegative & seq2 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) is nonnegative & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) holds seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = (seq1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) + (seq2 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ext-real ) ( ext-real ) set ) ) holds
( seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) is nonnegative & SUM seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = (SUM seq1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) + (SUM seq2 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ext-real ) ( ext-real ) set ) & Sum seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = (Sum seq1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) + (Sum seq2 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ext-real ) ( ext-real ) set ) ) ;

registration
let X be ( ( ) ( ) set ) ;
let F be ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ;
cluster non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total disjoint_valued for ( ( ) ( ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let F be ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ;
mode FinSequence of F -> ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like FinSequence-like ) FinSequence of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) means :: MEASURE8:def 1
 for k being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) st k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) in dom it : ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) : ( ( ) ( V73() V74() V75() V76() V77() V78() ) Element of bool NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) holds
it : ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) . k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) in F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

registration
let X be ( ( ) ( ) set ) ;
let F be ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ;
cluster Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like FinSequence-like disjoint_valued for ( ( ) ( ) FinSequence of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let F be ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ;
mode Sep_FinSequence of F is ( ( disjoint_valued ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like FinSequence-like disjoint_valued ) FinSequence of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let F be ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ;
mode Sep_Sequence of F is ( ( Function-like quasi_total disjoint_valued ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total disjoint_valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let F be ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ;
mode Set_Sequence of F -> ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) means :: MEASURE8:def 2
 for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) holds it : ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) in F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let A be ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ;
let F be ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ;
mode Covering of A,F -> ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Set_Sequence of F : ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) ) means :: MEASURE8:def 3
A : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) c= union (rng it : ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let F be ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ;
let FSets be ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Set_Sequence of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ;
let n be ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) ;
:: original: .
redefine func FSets . n -> ( ( ) ( ) Element of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let F be ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ;
let Sets be ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ;
mode Covering of Sets,F -> ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) means :: MEASURE8:def 4
 for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) holds it : ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) : ( ( ) ( Relation-like Function-like ) Element of Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) is ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Covering of Sets : ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let F be ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ;
let M be ( ( Function-like quasi_total zeroed nonnegative V102(X : ( ( ) ( ) set ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(X : ( ( ) ( ) set ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ;
let FSets be ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Set_Sequence of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ;
func vol (M,FSets) -> ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) means :: MEASURE8:def 5
 for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) holds it : ( ( ) ( ) set ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = M : ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) . (FSets : ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) ) : ( ( ) ( ) Element of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ;
end;

theorem :: MEASURE8:4
for X being ( ( ) ( ) set )
for F being ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of )
for M being ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) )
for FSets being ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Set_Sequence of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) holds vol (M : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ,FSets : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Set_Sequence of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) is nonnegative ;

definition
let X be ( ( ) ( ) set ) ;
let F be ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ;
let Sets be ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ;
let Cvr be ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Covering of Sets : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ;
let n be ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) ;
:: original: .
redefine func Cvr . n -> ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Covering of Sets : ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) . n : ( ( ) ( ) set ) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let F be ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ;
let Sets be ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ;
let M be ( ( Function-like quasi_total zeroed nonnegative V102(X : ( ( ) ( ) set ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(X : ( ( ) ( ) set ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ;
let Cvr be ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Covering of Sets : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ;
func Volume (M,Cvr) -> ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) means :: MEASURE8:def 6
 for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) holds it : ( ( ) ( ) Element of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = SUM (vol (M : ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) ,(Cvr : ( ( ) ( ) set ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) ) : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Covering of Sets : ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) . b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ;
end;

theorem :: MEASURE8:5
for X being ( ( ) ( ) set )
for F being ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of )
for M being ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) )
for Sets being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) )
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat)
for Cvr being ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Covering of Sets : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) holds 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural ext-real non positive non negative Function-like functional V56() real V58() V62() V73() V74() V75() V76() V77() V78() V79() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) <= (Volume (M : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ,Cvr : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Covering of b4 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ;

definition
let X be ( ( ) ( ) set ) ;
let F be ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ;
let M be ( ( Function-like quasi_total zeroed nonnegative V102(X : ( ( ) ( ) set ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(X : ( ( ) ( ) set ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ;
let A be ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ;
func Svc (M,A) -> ( ( ) ( V74() ) Subset of ( ( ) ( non empty ) set ) ) means :: MEASURE8:def 7
 for x being ( ( ) ( ext-real ) R_eal) holds
( x : ( ( ) ( ext-real ) R_eal) in it : ( ( ) ( ) set ) iff ex CA being ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Covering of A : ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) st x : ( ( ) ( ext-real ) R_eal) = SUM (vol (M : ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) ,CA : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Covering of A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) );
end;

registration
let X be ( ( ) ( ) set ) ;
let A be ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ;
let F be ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ;
let M be ( ( Function-like quasi_total zeroed nonnegative V102(X : ( ( ) ( ) set ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(X : ( ( ) ( ) set ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ;
cluster Svc (M : ( ( Function-like quasi_total zeroed nonnegative V102(X : ( ( ) ( ) set ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued zeroed nonnegative V102(X : ( ( ) ( ) set ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Element of bool [:F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ,A : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V74() ) Subset of ( ( ) ( non empty ) set ) ) -> non empty ;
end;

definition
let X be ( ( ) ( ) set ) ;
let F be ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ;
let M be ( ( Function-like quasi_total zeroed nonnegative V102(X : ( ( ) ( ) set ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(X : ( ( ) ( ) set ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ;
func C_Meas M -> ( ( Function-like quasi_total ) ( non empty Relation-like bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) Function of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V74() ) set ) ) means :: MEASURE8:def 8
 for A being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds it : ( ( Function-like quasi_total zeroed nonnegative V102(X : ( ( ) ( ) set ) ,M : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like M : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(M : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued zeroed nonnegative V102(X : ( ( ) ( ) set ) ,M : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Element of bool [:M : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = inf (Svc (M : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( V74() ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ;
end;

definition
func InvPairFunc -> ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty RAT : ( ( ) ( non empty V34() V73() V74() V75() V76() V79() ) set ) -valued INT : ( ( ) ( non empty V34() V73() V74() V75() V76() V77() V79() ) set ) -valued complex-valued ext-real-valued real-valued natural-valued ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,[:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty RAT : ( ( ) ( non empty V34() V73() V74() V75() V76() V79() ) set ) -valued INT : ( ( ) ( non empty V34() V73() V74() V75() V76() V77() V79() ) set ) -valued complex-valued ext-real-valued real-valued natural-valued ) set ) ) equals :: MEASURE8:def 9
PairFunc : ( ( Function-like quasi_total ) ( non empty Relation-like [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty RAT : ( ( ) ( non empty V34() V73() V74() V75() V76() V79() ) set ) -valued INT : ( ( ) ( non empty V34() V73() V74() V75() V76() V77() V79() ) set ) -valued complex-valued ext-real-valued real-valued natural-valued ) set ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27([:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty RAT : ( ( ) ( non empty V34() V73() V74() V75() V76() V79() ) set ) -valued INT : ( ( ) ( non empty V34() V73() V74() V75() V76() V77() V79() ) set ) -valued complex-valued ext-real-valued real-valued natural-valued ) set ) ) quasi_total complex-valued ext-real-valued real-valued natural-valued ) Element of bool [:[:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty RAT : ( ( ) ( non empty V34() V73() V74() V75() V76() V79() ) set ) -valued INT : ( ( ) ( non empty V34() V73() V74() V75() V76() V77() V79() ) set ) -valued complex-valued ext-real-valued real-valued natural-valued ) set ) ,NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty RAT : ( ( ) ( non empty V34() V73() V74() V75() V76() V79() ) set ) -valued INT : ( ( ) ( non empty V34() V73() V74() V75() V76() V77() V79() ) set ) -valued complex-valued ext-real-valued real-valued natural-valued ) set ) : ( ( ) ( non empty ) set ) ) " : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let F be ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ;
let Sets be ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ;
let Cvr be ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Covering of Sets : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ;
func On Cvr -> ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Covering of union (rng Sets : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) means :: MEASURE8:def 10
 for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) holds it : ( ( ) ( ) set ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) : ( ( ) ( ) Element of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) = (Cvr : ( ( Function-like quasi_total zeroed nonnegative V102(X : ( ( ) ( ) set ) ,Sets : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like Sets : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(Sets : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued zeroed nonnegative V102(X : ( ( ) ( ) set ) ,Sets : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Element of bool [:Sets : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . ((pr1 InvPairFunc : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty RAT : ( ( ) ( non empty V34() V73() V74() V75() V76() V79() ) set ) -valued INT : ( ( ) ( non empty V34() V73() V74() V75() V76() V77() V79() ) set ) -valued complex-valued ext-real-valued real-valued natural-valued ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,[:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty RAT : ( ( ) ( non empty V34() V73() V74() V75() V76() V79() ) set ) -valued INT : ( ( ) ( non empty V34() V73() V74() V75() V76() V77() V79() ) set ) -valued complex-valued ext-real-valued real-valued natural-valued ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total complex-valued ext-real-valued real-valued natural-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty RAT : ( ( ) ( non empty V34() V73() V74() V75() V76() V79() ) set ) -valued INT : ( ( ) ( non empty V34() V73() V74() V75() V76() V77() V79() ) set ) -valued complex-valued ext-real-valued real-valued natural-valued ) set ) : ( ( ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real V58() V62() V73() V74() V75() V76() V77() V78() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty ext-real Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Covering of Sets : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) . ((pr1 InvPairFunc : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty RAT : ( ( ) ( non empty V34() V73() V74() V75() V76() V79() ) set ) -valued INT : ( ( ) ( non empty V34() V73() V74() V75() V76() V77() V79() ) set ) -valued complex-valued ext-real-valued real-valued natural-valued ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,[:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty RAT : ( ( ) ( non empty V34() V73() V74() V75() V76() V79() ) set ) -valued INT : ( ( ) ( non empty V34() V73() V74() V75() V76() V77() V79() ) set ) -valued complex-valued ext-real-valued real-valued natural-valued ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total complex-valued ext-real-valued real-valued natural-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty RAT : ( ( ) ( non empty V34() V73() V74() V75() V76() V79() ) set ) -valued INT : ( ( ) ( non empty V34() V73() V74() V75() V76() V77() V79() ) set ) -valued complex-valued ext-real-valued real-valued natural-valued ) set ) : ( ( ) ( non empty ) set ) ) . b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real V58() V62() V73() V74() V75() V76() V77() V78() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . ((pr2 InvPairFunc : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty RAT : ( ( ) ( non empty V34() V73() V74() V75() V76() V79() ) set ) -valued INT : ( ( ) ( non empty V34() V73() V74() V75() V76() V77() V79() ) set ) -valued complex-valued ext-real-valued real-valued natural-valued ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,[:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty RAT : ( ( ) ( non empty V34() V73() V74() V75() V76() V79() ) set ) -valued INT : ( ( ) ( non empty V34() V73() V74() V75() V76() V77() V79() ) set ) -valued complex-valued ext-real-valued real-valued natural-valued ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total complex-valued ext-real-valued real-valued natural-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty RAT : ( ( ) ( non empty V34() V73() V74() V75() V76() V79() ) set ) -valued INT : ( ( ) ( non empty V34() V73() V74() V75() V76() V77() V79() ) set ) -valued complex-valued ext-real-valued real-valued natural-valued ) set ) : ( ( ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real V58() V62() V73() V74() V75() V76() V77() V78() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ;
end;

theorem :: MEASURE8:6
for X being ( ( ) ( ) set )
for F being ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of )
for M being ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) )
for k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real V58() V62() V73() V74() V75() V76() V77() V78() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ex m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) st
for Sets being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) )
for Cvr being ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Covering of Sets : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) holds (Partial_Sums (vol (M : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ,(On Cvr : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Covering of b6 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Covering of union (rng b6 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real V58() V62() V73() V74() V75() V76() V77() V78() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) <= (Partial_Sums (Volume (M : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ,Cvr : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Covering of b6 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ;

theorem :: MEASURE8:7
for X being ( ( ) ( ) set )
for F being ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of )
for M being ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) )
for Sets being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) )
for Cvr being ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Covering of Sets : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) holds inf (Svc (M : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ,(union (rng Sets : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( non empty V74() ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) <= SUM (Volume (M : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ,Cvr : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Covering of b4 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ;

theorem :: MEASURE8:8
for X being ( ( ) ( ) set )
for F being ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of )
for A being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) st A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) in F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) holds
(A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ,({} X : ( ( ) ( ) set ) ) : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural ext-real non positive non negative Function-like functional V56() real V73() V74() V75() V76() V77() V78() V79() ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) followed_by ({} X : ( ( ) ( ) set ) ) : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural ext-real non positive non negative Function-like functional V56() real V73() V74() V75() V76() V77() V78() V79() ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) is ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Covering of A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ;

theorem :: MEASURE8:9
for X being ( ( ) ( ) set )
for F being ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of )
for M being ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) )
for A being ( ( ) ( ) set ) st A : ( ( ) ( ) set ) in F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) holds
(C_Meas M : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) Function of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V74() ) set ) ) . A : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) <= M : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) . A : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ;

theorem :: MEASURE8:10
for X being ( ( ) ( ) set )
for F being ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of )
for M being ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) holds C_Meas M : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) Function of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V74() ) set ) ) is nonnegative ;

theorem :: MEASURE8:11
for X being ( ( ) ( ) set )
for F being ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of )
for M being ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) holds (C_Meas M : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) Function of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V74() ) set ) ) . {} : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural ext-real non positive non negative Function-like functional V56() real V73() V74() V75() V76() V77() V78() V79() ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural ext-real non positive non negative Function-like functional V56() real V58() V62() V73() V74() V75() V76() V77() V78() V79() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MEASURE8:12
for X being ( ( ) ( ) set )
for F being ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of )
for M being ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) )
for A, B being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) st A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) c= B : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds
(C_Meas M : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) Function of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V74() ) set ) ) . A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) <= (C_Meas M : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) Function of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V74() ) set ) ) . B : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ;

theorem :: MEASURE8:13
for X being ( ( ) ( ) set )
for F being ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of )
for M being ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) )
for Sets being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) holds (C_Meas M : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) Function of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V74() ) set ) ) . (union (rng Sets : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) <= SUM ((C_Meas M : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) Function of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V74() ) set ) ) * Sets : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ;

theorem :: MEASURE8:14
for X being ( ( ) ( ) set )
for F being ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of )
for M being ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) holds C_Meas M : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) Function of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V74() ) set ) ) is ( ( ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of X : ( ( ) ( ) set ) ) ;

definition
let X be ( ( ) ( ) set ) ;
let F be ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ;
let M be ( ( Function-like quasi_total zeroed nonnegative V102(X : ( ( ) ( ) set ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(X : ( ( ) ( ) set ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ;
:: original: C_Meas
redefine func C_Meas M -> ( ( ) ( non empty Relation-like bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of X : ( ( ) ( ) set ) ) ;
end;

begin

definition
let X be ( ( ) ( ) set ) ;
let F be ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ;
let M be ( ( Function-like quasi_total zeroed nonnegative V102(X : ( ( ) ( ) set ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(X : ( ( ) ( ) set ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ;
attr M is completely-additive means :: MEASURE8:def 11
 for FSets being ( ( Function-like quasi_total disjoint_valued ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total disjoint_valued ) Sep_Sequence of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) st union (rng FSets : ( ( Function-like quasi_total disjoint_valued ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total disjoint_valued ) Sep_Sequence of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) : ( ( ) ( ) Element of bool F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) in F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
SUM (M : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) * FSets : ( ( Function-like quasi_total disjoint_valued ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total disjoint_valued ) Sep_Sequence of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = M : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) . (union (rng FSets : ( ( Function-like quasi_total disjoint_valued ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total disjoint_valued ) Sep_Sequence of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) : ( ( ) ( ) Element of bool F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ;
end;

theorem :: MEASURE8:15
for X being ( ( ) ( ) set )
for F being ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of )
for FSets being ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Set_Sequence of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) holds Partial_Union FSets : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Set_Sequence of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) is ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Set_Sequence of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ;

theorem :: MEASURE8:16
for X being ( ( ) ( ) set )
for F being ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of )
for FSets being ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Set_Sequence of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) holds Partial_Diff_Union FSets : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Set_Sequence of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total disjoint_valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) is ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Set_Sequence of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ;

theorem :: MEASURE8:17
for X being ( ( ) ( ) set )
for F being ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of )
for A being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) )
for CA being ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Covering of A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) st A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) in F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) holds
ex FSets being ( ( Function-like quasi_total disjoint_valued ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total disjoint_valued ) Sep_Sequence of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) st
( A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) = union (rng FSets : ( ( Function-like quasi_total disjoint_valued ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total disjoint_valued ) Sep_Sequence of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) holds FSets : ( ( Function-like quasi_total disjoint_valued ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total disjoint_valued ) Sep_Sequence of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) c= CA : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Covering of b3 : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ) ;

theorem :: MEASURE8:18
for X being ( ( ) ( ) set )
for F being ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of )
for M being ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) st M : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) is completely-additive holds
for A being ( ( ) ( ) set ) st A : ( ( ) ( ) set ) in F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) holds
M : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) . A : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = (C_Meas M : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) : ( ( ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( ) ( ) set ) ) . A : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ;

theorem :: MEASURE8:19
for X being ( ( ) ( ) set )
for A being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) )
for C being ( ( ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of X : ( ( ) ( ) set ) ) st ( for B being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds (C : ( ( ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( ) ( ) set ) ) . (B : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) /\ A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) + (C : ( ( ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( ) ( ) set ) ) . (B : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) /\ (X : ( ( ) ( ) set ) \ A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ext-real ) ( ext-real ) set ) <= C : ( ( ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( ) ( ) set ) ) . B : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ) holds
A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) in sigma_Field C : ( ( ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( ) ( ) set ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MEASURE8:20
for X being ( ( ) ( ) set )
for F being ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of )
for M being ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) holds F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) c= sigma_Field (C_Meas M : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) : ( ( ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( ) ( ) set ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MEASURE8:21
for X being ( ( ) ( ) set )
for F being ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of )
for FSets being ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Set_Sequence of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) )
for M being ( ( Function-like quasi_total ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued ) Function of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) , ExtREAL : ( ( ) ( non empty V74() ) set ) ) holds M : ( ( Function-like quasi_total ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued ) Function of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) , ExtREAL : ( ( ) ( non empty V74() ) set ) ) * FSets : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Set_Sequence of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) ;

definition
let X be ( ( ) ( ) set ) ;
let F be ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ;
let FSets be ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Set_Sequence of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ;
let g be ( ( Function-like quasi_total ) ( non empty Relation-like F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued ) Function of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) , ExtREAL : ( ( ) ( non empty V74() ) set ) ) ;
:: original: *
redefine func g * FSets -> ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) ;
end;

theorem :: MEASURE8:22
for X being ( ( ) ( ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) )
for SSets being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) )
for M being ( ( Function-like quasi_total ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) quasi_total ext-real-valued ) Function of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V74() ) set ) ) holds M : ( ( Function-like quasi_total ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) quasi_total ext-real-valued ) Function of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V74() ) set ) ) * SSets : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) ;

definition
let X be ( ( ) ( ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ;
let SSets be ( ( S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) -valued Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) -valued bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ) ;
let g be ( ( Function-like quasi_total ) ( non empty Relation-like S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ) quasi_total ext-real-valued ) Function of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V74() ) set ) ) ;
:: original: *
redefine func g * SSets -> ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) ;
end;

theorem :: MEASURE8:23
for F, G being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V74() ) set ) )
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) st ( for m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) st m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) <= n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) holds
F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V74() ) set ) ) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) <= G : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V74() ) set ) ) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ) holds
(Ser F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V74() ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) <= (Ser G : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V74() ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ;

theorem :: MEASURE8:24
for X being ( ( ) ( ) set )
for C being ( ( ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of X : ( ( ) ( ) set ) )
for seq being ( ( Function-like quasi_total disjoint_valued ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined sigma_Field b2 : ( ( ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( ) ( ) set ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total disjoint_valued ) Sep_Sequence of (sigma_Field C : ( ( ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( ) ( ) set ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) holds
( union (rng seq : ( ( Function-like quasi_total disjoint_valued ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined sigma_Field b2 : ( ( ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( ) ( ) set ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total disjoint_valued ) Sep_Sequence of (sigma_Field b2 : ( ( ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( ) ( ) set ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) in sigma_Field C : ( ( ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( ) ( ) set ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & C : ( ( ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( ) ( ) set ) ) . (union (rng seq : ( ( Function-like quasi_total disjoint_valued ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined sigma_Field b2 : ( ( ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( ) ( ) set ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total disjoint_valued ) Sep_Sequence of (sigma_Field b2 : ( ( ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( ) ( ) set ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = Sum (C : ( ( ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( ) ( ) set ) ) * seq : ( ( Function-like quasi_total disjoint_valued ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined sigma_Field b2 : ( ( ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( ) ( ) set ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total disjoint_valued ) Sep_Sequence of (sigma_Field b2 : ( ( ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( ) ( ) set ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ) ;

theorem :: MEASURE8:25
for X being ( ( ) ( ) set )
for C being ( ( ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of X : ( ( ) ( ) set ) )
for seq being ( ( sigma_Field b2 : ( ( ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( ) ( ) set ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined sigma_Field b2 : ( ( ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( ) ( ) set ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of sigma_Field C : ( ( ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( ) ( ) set ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) holds Union seq : ( ( sigma_Field b2 : ( ( ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( ) ( ) set ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined sigma_Field b2 : ( ( ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( ) ( ) set ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of sigma_Field b2 : ( ( ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( ) ( ) set ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) in sigma_Field C : ( ( ) ( non empty Relation-like bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( ) ( ) set ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MEASURE8:26
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for SSets being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st SSets : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) is non-descending holds
lim (M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) * SSets : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (lim SSets : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ;

theorem :: MEASURE8:27
for X being ( ( ) ( ) set )
for F being ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of )
for M being ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) )
for FSets being ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Set_Sequence of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) st FSets : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Set_Sequence of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) is non-descending holds
M : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) * FSets : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Set_Sequence of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) is non-decreasing ;

theorem :: MEASURE8:28
for X being ( ( ) ( ) set )
for F being ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of )
for M being ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) )
for FSets being ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Set_Sequence of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) st FSets : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Set_Sequence of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) is non-ascending holds
M : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) * FSets : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Set_Sequence of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) is non-increasing ;

theorem :: MEASURE8:29
for X being ( ( ) ( ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) )
for M being ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) )
for SSets being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) st SSets : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) is non-descending holds
M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) * SSets : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) is non-decreasing ;

theorem :: MEASURE8:30
for X being ( ( ) ( ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) )
for M being ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) )
for SSets being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) st SSets : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) is non-ascending holds
M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) * SSets : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) is non-increasing ;

theorem :: MEASURE8:31
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for SSets being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st SSets : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) is non-ascending & M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (SSets : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural ext-real non positive non negative Function-like functional V56() real V58() V62() V73() V74() V75() V76() V77() V78() V79() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) M7(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) < +infty : ( ( ) ( non empty ext-real positive non negative non real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) holds
lim (M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) * SSets : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (lim SSets : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ;

definition
let X be ( ( ) ( ) set ) ;
let F be ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ;
let m be ( ( Function-like quasi_total zeroed nonnegative V102(X : ( ( ) ( ) set ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(X : ( ( ) ( ) set ) ,F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ;
let M be ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ) quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ) ;
pred M is_extension_of m means :: MEASURE8:def 12
 for A being ( ( ) ( ) set ) st A : ( ( ) ( ) set ) in F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
M : ( ( ) ( ) set ) . A : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = m : ( ( Function-like quasi_total zeroed nonnegative V102(X : ( ( ) ( ) set ) ,S : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like S : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(S : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued zeroed nonnegative V102(X : ( ( ) ( ) set ) ,S : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Element of bool [:S : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . A : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ;
end;

theorem :: MEASURE8:32
for X being ( ( non empty ) ( non empty ) set )
for F being ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of )
for m being ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) st ex M being ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like sigma b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( sigma b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of sigma F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) st M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like sigma b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( sigma b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of sigma b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) is_extension_of m : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) holds
m : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) is completely-additive ;

theorem :: MEASURE8:33
for X being ( ( non empty ) ( non empty ) set )
for F being ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of )
for m being ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) st m : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) is completely-additive holds
ex M being ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like sigma b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( sigma b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of sigma F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) st
( M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like sigma b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( sigma b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of sigma b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) is_extension_of m : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) & M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like sigma b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( sigma b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of sigma b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) = (sigma_Meas (C_Meas m : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) : ( ( ) ( non empty Relation-like bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like sigma_Field (C_Meas b3 : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) : ( ( ) ( non empty Relation-like bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( sigma_Field (C_Meas b3 : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) : ( ( ) ( non empty Relation-like bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued zeroed nonnegative sigma-additive ) Element of bool [:(sigma_Field (C_Meas b3 : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) : ( ( ) ( non empty Relation-like bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) | (sigma F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like sigma_Field (C_Meas b3 : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) : ( ( ) ( non empty Relation-like bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:(sigma_Field (C_Meas b3 : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) : ( ( ) ( non empty Relation-like bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MEASURE8:34
for X being ( ( ) ( ) set )
for F being ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of )
for M being ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) )
for k being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat)
for FSets being ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Set_Sequence of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) holds M : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) . (FSets : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Set_Sequence of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) < +infty : ( ( ) ( non empty ext-real positive non negative non real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ) holds
M : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) . ((Partial_Union FSets : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Set_Sequence of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) . k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) < +infty : ( ( ) ( non empty ext-real positive non negative non real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ;

theorem :: MEASURE8:35
for X being ( ( non empty ) ( non empty ) set )
for F being ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of )
for m being ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) st m : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) is completely-additive & ex Aseq being ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V34() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ) Set_Sequence of F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) st
( ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) holds m : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) . (Aseq : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like sigma b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( sigma b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of sigma b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural ext-real non negative V56() real ) Nat) ) : ( ( ) ( ext-real ) Element of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) < +infty : ( ( ) ( non empty ext-real positive non negative non real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ) & X : ( ( non empty ) ( non empty ) set ) = union (rng Aseq : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like sigma b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( sigma b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of sigma b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V74() ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) holds
for M being ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like sigma b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( sigma b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of sigma F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) st M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like sigma b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( sigma b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of sigma b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) is_extension_of m : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) holds
M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like sigma b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( sigma b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of sigma b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) = (sigma_Meas (C_Meas m : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) : ( ( ) ( non empty Relation-like bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like sigma_Field (C_Meas b3 : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) : ( ( ) ( non empty Relation-like bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( sigma_Field (C_Meas b3 : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) : ( ( ) ( non empty Relation-like bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued zeroed nonnegative sigma-additive ) Element of bool [:(sigma_Field (C_Meas b3 : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) : ( ( ) ( non empty Relation-like bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) | (sigma F : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like sigma_Field (C_Meas b3 : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) : ( ( ) ( non empty Relation-like bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:(sigma_Field (C_Meas b3 : ( ( Function-like quasi_total zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) ( non empty Relation-like b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) quasi_total ext-real-valued zeroed nonnegative V102(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) Measure of b2 : ( ( non empty compl-closed V85() ) ( non empty compl-closed V84() V85() V86() ) Field_Subset of ) ) ) : ( ( ) ( non empty Relation-like bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like V27( bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) quasi_total ext-real-valued ) C_Measure of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V84() V85() V86() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ;