begin
definition
let X,
Y be ( ( ) ( )
set ) ;
let F be ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
X : ( ( ) ( )
set ) ,
Y : ( ( ) ( )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
X : ( ( ) ( )
set ) ,
Y : ( ( ) ( )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
X : ( ( ) ( )
set ) ,
Y : ( ( ) ( )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( ) ( )
set ) ) ;
let D be ( ( ) ( )
set ) ;
func F || D -> ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ,
Y : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ,
Y : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ,
Y : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) )
means
for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
it : ( ( ) ( )
Element of
Y : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) )
. n : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) : ( (
Function-like ) (
Relation-like X : ( (
V41()
right_end ) (
V41()
right_end )
set )
-defined Y : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like )
Element of
K6(
K7(
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ,
Y : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= (F : ( ( Function-like V51(Y : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) V57() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( V57() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V41() ) set ) ) V66() nonnegative sigma-additive ) ( non empty Relation-like Y : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) V57() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( V57() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V41() ) set ) -valued Function-like total V51(Y : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) V57() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( V57() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V41() ) set ) ) V58() V66() nonnegative sigma-additive ) Element of K6(K7(Y : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) V57() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( V57() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V41() ) set ) ) : ( ( ) ( V58() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real ) Nat) ) : ( (
Function-like ) (
Relation-like X : ( (
V41()
right_end ) (
V41()
right_end )
set )
-defined Y : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like ext-real )
Element of
K6(
K7(
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ,
Y : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
| D : ( (
Function-like ) (
Relation-like X : ( (
V41()
right_end ) (
V41()
right_end )
set )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like V58() )
Element of
K6(
K7(
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) (
V58() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like X : ( (
V41()
right_end ) (
V41()
right_end )
set )
-defined Y : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like )
Element of
K6(
K7(
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ,
Y : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
end;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) )
for
x being ( ( ) ( )
Element of
X : ( ( non
empty ) ( non
empty )
set ) )
for
D being ( ( ) ( )
set ) st
x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) )
in D : ( ( ) ( )
set ) &
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
# x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
V57()
V58()
V59() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) is
convergent holds
(F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) || D : ( ( ) ( ) set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
# x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
V57()
V58()
V59() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) is
convergent ;
theorem
for
X,
Y,
D being ( ( ) ( )
set )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( ) ( )
set ) ,
b2 : ( ( ) ( )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( ) ( )
set ) ,
b2 : ( ( ) ( )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( ) ( )
set ) ,
b2 : ( ( ) ( )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( ) ( )
set ) ) st
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( ) ( )
set ) ,
b2 : ( ( ) ( )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( ) ( )
set ) ,
b2 : ( ( ) ( )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( ) ( )
set ) ,
b2 : ( ( ) ( )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( ) ( )
set ) ) is
with_the_same_dom holds
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( ) ( )
set ) ,
b2 : ( ( ) ( )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( ) ( )
set ) ,
b2 : ( ( ) ( )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( ) ( )
set ) ,
b2 : ( ( ) ( )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( ) ( )
set ) )
|| D : ( ( ) ( )
set ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( ) ( )
set ) ,
b2 : ( ( ) ( )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( ) ( )
set ) ,
b2 : ( ( ) ( )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( ) ( )
set ) ,
b2 : ( ( ) ( )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( ) ( )
set ) ) is
with_the_same_dom ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) )
for
D being ( ( ) ( )
set ) st
D : ( ( ) ( )
set )
c= dom (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V40() V41() V42() V43() V44() V45() V46() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) ( )
Element of
X : ( ( non
empty ) ( non
empty )
set ) ) st
x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) )
in D : ( ( ) ( )
set ) holds
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
# x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
V57()
V58()
V59() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) is
convergent ) holds
(lim F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like V58() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) (
V58() )
set ) ) : ( ( ) ( )
set ) )
| D : ( ( ) ( )
set ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like V58() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) (
V58() )
set ) ) : ( ( ) ( )
set ) )
= lim (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) || D : ( ( ) ( ) set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like V58() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) (
V58() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
S being ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
X : ( ( non
empty ) ( non
empty )
set ) )
for
E being ( ( ) ( )
Element of
S : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) )
for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) st
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) is
with_the_same_dom &
E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
c= dom (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V40() V41() V42() V43() V44() V45() V46() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) & ( for
m being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
. m : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) )
is_measurable_on E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ) holds
(F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) || E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
. n : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) )
is_measurable_on E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ;
theorem
for
seq being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
V57()
V58()
V59() )
Real_Sequence) holds
Partial_Sums (R_EAL seq : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) V57() V58() V59() ) Real_Sequence) ) : ( (
Function-like ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) (
V58() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) (
V58() )
set ) ) : ( ( ) ( )
set ) )
= R_EAL (Partial_Sums seq : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) V57() V58() V59() ) Real_Sequence) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
V57()
V58()
V59() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) (
V58() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
S being ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
X : ( ( non
empty ) ( non
empty )
set ) )
for
E being ( ( ) ( )
Element of
S : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) ) st ( for
x being ( ( ) ( )
Element of
X : ( ( non
empty ) ( non
empty )
set ) ) st
x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) )
in E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) holds
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
# x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
V57()
V58()
V59() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) is
summable ) holds
for
x being ( ( ) ( )
Element of
X : ( ( non
empty ) ( non
empty )
set ) ) st
x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) )
in E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) holds
(F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) || E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
# x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
V57()
V58()
V59() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) is
summable ;
definition
let X be ( ( non
empty ) ( non
empty )
set ) ;
let F be ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
X : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
X : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
X : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) ) ;
func Partial_Sums F -> ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) )
means
(
it : ( (
Function-like V51(
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
Element of
K6(
K7(
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) (
V58() )
set ) ) : ( ( ) ( )
set ) )
. 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non
positive non
negative V40()
V41()
V42()
V43()
V44()
V45()
V46()
V72()
V74() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
Function-like ) (
Relation-like X : ( (
V41()
right_end ) (
V41()
right_end )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like ext-real V57()
V58()
V59() )
Element of
K6(
K7(
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) )
= F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
. 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non
positive non
negative V40()
V41()
V42()
V43()
V44()
V45()
V46()
V72()
V74() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
Function-like ) (
complex Relation-like X : ( (
V41()
right_end ) (
V41()
right_end )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) & ( for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
it : ( (
Function-like V51(
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
Element of
K6(
K7(
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) (
V58() )
set ) ) : ( ( ) ( )
set ) )
. (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real ) Nat) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V40() V41() V42() V43() V44() V45() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real V40()
V41()
V42()
V43()
V44()
V45()
V72()
V74() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
Function-like ) (
Relation-like X : ( (
V41()
right_end ) (
V41()
right_end )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like ext-real V57()
V58()
V59() )
Element of
K6(
K7(
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) )
= (it : ( ( Function-like V51(F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) V57() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( V57() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V41() ) set ) ) V66() nonnegative sigma-additive ) ( non empty Relation-like F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) V57() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( V57() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V41() ) set ) -valued Function-like total V51(F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) V57() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( V57() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V41() ) set ) ) V58() V66() nonnegative sigma-additive ) Element of K6(K7(F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) V57() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( V57() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V41() ) set ) ) : ( ( ) ( V58() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real ) Nat) ) : ( (
Function-like ) (
Relation-like X : ( (
V41()
right_end ) (
V41()
right_end )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like ext-real V57()
V58()
V59() )
Element of
K6(
K7(
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) )
+ (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) V57() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( V57() ) set ) ) : ( ( ) ( ) set ) ) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real ) Nat) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V40() V41() V42() V43() V44() V45() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V40() V41() V42() V43() V44() V45() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
complex Relation-like X : ( (
V41()
right_end ) (
V41()
right_end )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like X : ( (
V41()
right_end ) (
V41()
right_end )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) ) );
end;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) ) holds
Partial_Sums (R_EAL F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V41() ) set ) )) : ( ( ) ( non
empty functional )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V41() ) set ) )) : ( ( ) ( non
empty functional )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= R_EAL (Partial_Sums F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V41() ) set ) )) : ( ( ) ( non
empty functional )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) )
for
n,
m being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat)
for
z being ( ( ) ( )
set ) st
z : ( ( ) ( )
set )
in dom ((Partial_Sums F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real ) Nat) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) &
m : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat)
<= n : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
(
z : ( ( ) ( )
set )
in dom ((Partial_Sums F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real ) Nat) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) &
z : ( ( ) ( )
set )
in dom (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real ) Nat) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) ) holds
R_EAL F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V41() ) set ) )) : ( ( ) ( non
empty functional )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) is
additive ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) )
for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
dom ((Partial_Sums F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real ) Nat) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
= meet { (dom (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V40() V41() V42() V43() V44() V45() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) -valued Function-like V57() V58() V59() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( V57() V58() V59() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) where k is ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V40() V41() V42() V43() V44() V45() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) : k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V40() V41() V42() V43() V44() V45() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real ) Nat) } : ( ( ) ( )
set ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) )
for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) st
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) is
with_the_same_dom holds
dom ((Partial_Sums F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real ) Nat) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
= dom (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V40() V41() V42() V43() V44() V45() V46() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) )
for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat)
for
x being ( ( ) ( )
Element of
X : ( ( non
empty ) ( non
empty )
set ) )
for
D being ( ( ) ( )
set ) st
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) is
with_the_same_dom &
D : ( ( ) ( )
set )
c= dom (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V40() V41() V42() V43() V44() V45() V46() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) &
x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) )
in D : ( ( ) ( )
set ) holds
(Partial_Sums (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) V57() V58() V59() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( V57() V58() V59() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
V57()
V58()
V59() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) )
. n : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) : ( ( ) (
complex real ext-real )
Element of
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
= ((Partial_Sums F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
V57()
V58()
V59() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) )
. n : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) : ( ( ) (
complex real ext-real )
Element of
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) )
for
x being ( ( ) ( )
Element of
X : ( ( non
empty ) ( non
empty )
set ) )
for
D being ( ( ) ( )
set ) st
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) is
with_the_same_dom &
D : ( ( ) ( )
set )
c= dom (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V40() V41() V42() V43() V44() V45() V46() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) &
x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) )
in D : ( ( ) ( )
set ) holds
(
Partial_Sums (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
V57()
V58()
V59() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
V57()
V58()
V59() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) is
convergent iff
(Partial_Sums F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
# x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
V57()
V58()
V59() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) is
convergent ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) )
for
f being ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
PartFunc of ,)
for
x being ( ( ) ( )
Element of
X : ( ( non
empty ) ( non
empty )
set ) ) st
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) is
with_the_same_dom &
dom f : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
PartFunc of ,) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
c= dom (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V40() V41() V42() V43() V44() V45() V46() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) &
x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) )
in dom f : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
PartFunc of ,) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
PartFunc of ,)
. x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) (
complex real ext-real )
Element of
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
= Sum (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
V57()
V58()
V59() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
complex real ext-real )
Element of
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) holds
f : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
PartFunc of ,)
. x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) (
complex real ext-real )
Element of
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
= lim ((Partial_Sums F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
V57()
V58()
V59() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
complex real ext-real )
Element of
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
S being ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
X : ( ( non
empty ) ( non
empty )
set ) )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) )
for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) st ( for
m being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
. m : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) )
is_simple_func_in S : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) holds
(Partial_Sums F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
. n : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) )
is_simple_func_in S : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
S being ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
X : ( ( non
empty ) ( non
empty )
set ) )
for
E being ( ( ) ( )
Element of
S : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) )
for
m being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) st ( for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
. n : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) )
is_measurable_on E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ) holds
(Partial_Sums F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
. m : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) )
is_measurable_on E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) ) st
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) is
with_the_same_dom holds
Partial_Sums F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) is
with_the_same_dom ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
S being ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
X : ( ( non
empty ) ( non
empty )
set ) )
for
E being ( ( ) ( )
Element of
S : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) ) st
dom (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V40() V41() V42() V43() V44() V45() V46() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
= E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) &
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) is
with_the_same_dom & ( for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
(Partial_Sums F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
. n : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) )
is_measurable_on E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ) & ( for
x being ( ( ) ( )
Element of
X : ( ( non
empty ) ( non
empty )
set ) ) st
x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) )
in E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) holds
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
# x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
V57()
V58()
V59() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) is
summable ) holds
lim (Partial_Sums F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like V58() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) (
V58() )
set ) ) : ( ( ) ( )
set ) )
is_measurable_on E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
S being ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
X : ( ( non
empty ) ( non
empty )
set ) )
for
M being ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
S : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) ) st ( for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
. n : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) )
is_integrable_on M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ) holds
for
m being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
(Partial_Sums F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
. m : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) )
is_integrable_on M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ;
begin
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
f being ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
PartFunc of ,)
for
A being ( ( ) ( )
set ) holds
(
(Re f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like V57() ) PartFunc of ,) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) )
| A : ( ( ) ( )
set ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) )
= Re (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like V57() ) PartFunc of ,) | A : ( ( ) ( ) set ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) &
(Im f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like V57() ) PartFunc of ,) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) )
| A : ( ( ) ( )
set ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) )
= Im (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like V57() ) PartFunc of ,) | A : ( ( ) ( ) set ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
D being ( ( ) ( )
set )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) ) holds
Re (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) || D : ( ( ) ( ) set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) )) : ( ( ) ( non
empty functional )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= (Re F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) )) : ( ( ) ( non
empty functional )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
|| D : ( ( ) ( )
set ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
D being ( ( ) ( )
set )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) ) holds
Im (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) || D : ( ( ) ( ) set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) )) : ( ( ) ( non
empty functional )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= (Im F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) )) : ( ( ) ( non
empty functional )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
|| D : ( ( ) ( )
set ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
x being ( ( ) ( )
Element of
X : ( ( non
empty ) ( non
empty )
set ) )
for
D being ( ( ) ( )
set )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) ) st
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) is
with_the_same_dom &
D : ( ( ) ( )
set )
c= dom (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V40() V41() V42() V43() V44() V45() V46() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) &
x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) )
in D : ( ( ) ( )
set ) &
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
# x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) is
convergent holds
(F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) || D : ( ( ) ( ) set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
# x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) is
convergent ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) ) holds
(
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) is
with_the_same_dom iff
Re F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) )) : ( ( ) ( non
empty functional )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) is
with_the_same_dom ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) ) holds
(
Re F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) )) : ( ( ) ( non
empty functional )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) is
with_the_same_dom iff
Im F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) )) : ( ( ) ( non
empty functional )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) is
with_the_same_dom ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
D being ( ( ) ( )
set )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) ) st
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) is
with_the_same_dom &
D : ( ( ) ( )
set )
= dom (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V40() V41() V42() V43() V44() V45() V46() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) ( )
Element of
X : ( ( non
empty ) ( non
empty )
set ) ) st
x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) )
in D : ( ( ) ( )
set ) holds
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
# x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) is
convergent ) holds
(lim F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
| D : ( ( ) ( )
set ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
= lim (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) || D : ( ( ) ( ) set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
S being ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
X : ( ( non
empty ) ( non
empty )
set ) )
for
E being ( ( ) ( )
Element of
S : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat)
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) ) st
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) is
with_the_same_dom &
E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
c= dom (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V40() V41() V42() V43() V44() V45() V46() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) & ( for
m being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
. m : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
is_measurable_on E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ) holds
(F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) || E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
. n : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
is_measurable_on E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
S being ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
X : ( ( non
empty ) ( non
empty )
set ) )
for
E being ( ( ) ( )
Element of
S : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) ) st
E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
c= dom (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V40() V41() V42() V43() V44() V45() V46() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) &
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) is
with_the_same_dom & ( for
x being ( ( ) ( )
Element of
X : ( ( non
empty ) ( non
empty )
set ) ) st
x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) )
in E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) holds
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
# x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) is
summable ) holds
for
x being ( ( ) ( )
Element of
X : ( ( non
empty ) ( non
empty )
set ) ) st
x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) )
in E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) holds
(F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) || E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
# x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) is
summable ;
definition
let X be ( ( non
empty ) ( non
empty )
set ) ;
let F be ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
X : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
X : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
X : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) ) ;
func Partial_Sums F -> ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) )
means
(
it : ( (
Function-like V51(
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
Element of
K6(
K7(
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) (
V58() )
set ) ) : ( ( ) ( )
set ) )
. 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non
positive non
negative V40()
V41()
V42()
V43()
V44()
V45()
V46()
V72()
V74() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
Function-like ) (
Relation-like X : ( (
V41()
right_end ) (
V41()
right_end )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like ext-real V57() )
Element of
K6(
K7(
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
= F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
. 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non
positive non
negative V40()
V41()
V42()
V43()
V44()
V45()
V46()
V72()
V74() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
Function-like ) (
complex Relation-like X : ( (
V41()
right_end ) (
V41()
right_end )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) & ( for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
it : ( (
Function-like V51(
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
Element of
K6(
K7(
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) (
V58() )
set ) ) : ( ( ) ( )
set ) )
. (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real ) Nat) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V40() V41() V42() V43() V44() V45() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real V40()
V41()
V42()
V43()
V44()
V45()
V72()
V74() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
Function-like ) (
Relation-like X : ( (
V41()
right_end ) (
V41()
right_end )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like ext-real V57() )
Element of
K6(
K7(
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
= (it : ( ( Function-like V51(F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) V57() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( V57() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V41() ) set ) ) V66() nonnegative sigma-additive ) ( non empty Relation-like F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) V57() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( V57() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V41() ) set ) -valued Function-like total V51(F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) V57() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( V57() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V41() ) set ) ) V58() V66() nonnegative sigma-additive ) Element of K6(K7(F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) V57() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( V57() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V41() ) set ) ) : ( ( ) ( V58() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real ) Nat) ) : ( (
Function-like ) (
Relation-like X : ( (
V41()
right_end ) (
V41()
right_end )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like ext-real V57() )
Element of
K6(
K7(
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
+ (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) V57() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( V57() ) set ) ) : ( ( ) ( ) set ) ) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real ) Nat) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V40() V41() V42() V43() V44() V45() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V40() V41() V42() V43() V44() V45() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
complex Relation-like X : ( (
V41()
right_end ) (
V41()
right_end )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like X : ( (
V41()
right_end ) (
V41()
right_end )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) ) );
end;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) ) holds
(
Partial_Sums (Re F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) )) : ( ( ) ( non
empty functional )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
= Re (Partial_Sums F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) )) : ( ( ) ( non
empty functional )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) &
Partial_Sums (Im F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) )) : ( ( ) ( non
empty functional )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
= Im (Partial_Sums F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) )) : ( ( ) ( non
empty functional )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
n,
m being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat)
for
z being ( ( ) ( )
set )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) ) st
z : ( ( ) ( )
set )
in dom ((Partial_Sums F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real ) Nat) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) &
m : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat)
<= n : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
(
z : ( ( ) ( )
set )
in dom ((Partial_Sums F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real ) Nat) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) &
z : ( ( ) ( )
set )
in dom (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real ) Nat) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat)
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) ) holds
dom ((Partial_Sums F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real ) Nat) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
= meet { (dom (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V40() V41() V42() V43() V44() V45() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like V57() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( V57() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) where k is ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V40() V41() V42() V43() V44() V45() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) : k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V40() V41() V42() V43() V44() V45() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real ) Nat) } : ( ( ) ( )
set ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat)
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) ) st
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) is
with_the_same_dom holds
dom ((Partial_Sums F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real ) Nat) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
= dom (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V40() V41() V42() V43() V44() V45() V46() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat)
for
x being ( ( ) ( )
Element of
X : ( ( non
empty ) ( non
empty )
set ) )
for
D being ( ( ) ( )
set )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) ) st
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) is
with_the_same_dom &
D : ( ( ) ( )
set )
c= dom (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V40() V41() V42() V43() V44() V45() V46() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) &
x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) )
in D : ( ( ) ( )
set ) holds
(Partial_Sums (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) V57() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( V57() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
. n : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
= ((Partial_Sums F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
. n : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) ) st
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) is
with_the_same_dom holds
Partial_Sums F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) is
with_the_same_dom ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
x being ( ( ) ( )
Element of
X : ( ( non
empty ) ( non
empty )
set ) )
for
D being ( ( ) ( )
set )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) ) st
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) is
with_the_same_dom &
D : ( ( ) ( )
set )
c= dom (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V40() V41() V42() V43() V44() V45() V46() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) &
x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) )
in D : ( ( ) ( )
set ) holds
(
Partial_Sums (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) is
convergent iff
(Partial_Sums F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
# x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) is
convergent ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
x being ( ( ) ( )
Element of
X : ( ( non
empty ) ( non
empty )
set ) )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) )
for
f being ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
PartFunc of ,) st
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) is
with_the_same_dom &
dom f : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
PartFunc of ,) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
c= dom (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V40() V41() V42() V43() V44() V45() V46() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) &
x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) )
in dom f : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
PartFunc of ,) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) &
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
# x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) is
summable &
f : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
PartFunc of ,)
. x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) (
complex )
set )
= Sum (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) holds
f : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
PartFunc of ,)
. x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) (
complex )
set )
= lim ((Partial_Sums F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
S being ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
X : ( ( non
empty ) ( non
empty )
set ) )
for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat)
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) ) st ( for
m being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
. m : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
is_simple_func_in S : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) holds
(Partial_Sums F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
. n : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
is_simple_func_in S : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
S being ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
X : ( ( non
empty ) ( non
empty )
set ) )
for
E being ( ( ) ( )
Element of
S : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
for
m being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat)
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) ) st ( for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
. n : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
is_measurable_on E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ) holds
(Partial_Sums F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
. m : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
is_measurable_on E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
S being ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
X : ( ( non
empty ) ( non
empty )
set ) )
for
E being ( ( ) ( )
Element of
S : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) ) st
dom (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V40() V41() V42() V43() V44() V45() V46() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
= E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) &
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) is
with_the_same_dom & ( for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
(Partial_Sums F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
. n : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
is_measurable_on E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ) & ( for
x being ( ( ) ( )
Element of
X : ( ( non
empty ) ( non
empty )
set ) ) st
x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) )
in E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) holds
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
# x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) is
summable ) holds
lim (Partial_Sums F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
is_measurable_on E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
S being ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
X : ( ( non
empty ) ( non
empty )
set ) )
for
M being ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
S : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) ) st ( for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
. n : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
is_integrable_on M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ) holds
for
m being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
(Partial_Sums F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
. m : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
is_integrable_on M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ;
begin
begin
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
S being ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
X : ( ( non
empty ) ( non
empty )
set ) )
for
M being ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
S : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
for
E being ( ( ) ( )
Element of
S : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) )
for
P being ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like V58() )
PartFunc of ,) st
E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
= dom (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V41() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V41() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V41() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V40() V41() V42() V43() V44() V45() V46() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like V58() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) (
V58() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) &
E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
= dom P : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like V58() )
PartFunc of ,) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) & ( for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
. n : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like V58() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) (
V58() )
set ) ) : ( ( ) ( )
set ) )
is_measurable_on E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ) &
P : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like V58() )
PartFunc of ,)
is_integrable_on M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) & ( for
x being ( ( ) ( )
Element of
X : ( ( non
empty ) ( non
empty )
set ) )
for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) st
x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) )
in E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) holds
|.(F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V41() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V41() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V41() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real ) Nat) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V41() ) set ) -valued Function-like V58() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V41() ) set ) ) : ( ( ) ( V58() ) set ) ) : ( ( ) ( ) set ) ) .| : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like V58() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) (
V58() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) (
ext-real )
Element of
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
<= P : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like V58() )
PartFunc of ,)
. x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) (
ext-real )
Element of
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) ) & ( for
x being ( ( ) ( )
Element of
X : ( ( non
empty ) ( non
empty )
set ) ) st
x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) )
in E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) holds
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
# x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) (
V58() )
set ) ) : ( ( ) ( )
set ) ) is
convergent ) holds
lim F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like V58() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) (
V58() )
set ) ) : ( ( ) ( )
set ) )
is_integrable_on M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
S being ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
X : ( ( non
empty ) ( non
empty )
set ) )
for
M being ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
S : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
for
E being ( ( ) ( )
Element of
S : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) )
for
P being ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
PartFunc of ,) st
E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
= dom (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V40() V41() V42() V43() V44() V45() V46() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) &
E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
= dom P : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
PartFunc of ,) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) & ( for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
. n : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) )
is_measurable_on E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ) &
P : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
PartFunc of ,)
is_integrable_on M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) & ( for
x being ( ( ) ( )
Element of
X : ( ( non
empty ) ( non
empty )
set ) )
for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) st
x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) )
in E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) holds
|.(F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real ) Nat) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) -valued Function-like V57() V58() V59() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( V57() V58() V59() ) set ) ) : ( ( ) ( ) set ) ) .| : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) (
complex real ext-real )
Element of
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
<= P : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
PartFunc of ,)
. x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) (
complex real ext-real )
Element of
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) ) & ( for
x being ( ( ) ( )
Element of
X : ( ( non
empty ) ( non
empty )
set ) ) st
x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) )
in E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) holds
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
# x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
V57()
V58()
V59() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) is
convergent ) holds
lim F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like V58() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) (
V58() )
set ) ) : ( ( ) ( )
set ) )
is_integrable_on M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
S being ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
X : ( ( non
empty ) ( non
empty )
set ) )
for
M being ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
S : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
for
E being ( ( ) ( )
Element of
S : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) )
for
P being ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
PartFunc of ,) st
E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
= dom (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V40() V41() V42() V43() V44() V45() V46() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) &
E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
= dom P : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
PartFunc of ,) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) & ( for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
. n : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
V57()
V58()
V59() )
Real_Sequence) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) )
is_measurable_on E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ) &
P : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
PartFunc of ,)
is_integrable_on M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) & ( for
x being ( ( ) ( )
Element of
X : ( ( non
empty ) ( non
empty )
set ) )
for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) st
x : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
V57()
V58()
V59() )
Real_Sequence)
in E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) holds
|.(F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) -valued Function-like V57() V58() V59() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( V57() V58() V59() ) set ) ) : ( ( ) ( ) set ) ) .| : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) )
. x : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
V57()
V58()
V59() )
Real_Sequence) : ( ( ) (
complex real ext-real )
Element of
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
<= P : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
PartFunc of ,)
. x : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
V57()
V58()
V59() )
Real_Sequence) : ( ( ) (
complex real ext-real )
Element of
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) ) holds
ex
I being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
V57()
V58()
V59() )
Real_Sequence) st
( ( for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
I : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
V57()
V58()
V59() )
Real_Sequence)
. n : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) (
complex real ext-real )
Element of
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
= Integral (
M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ,
(F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
ext-real )
Element of
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) ) & ( ( for
x being ( ( ) ( )
Element of
X : ( ( non
empty ) ( non
empty )
set ) ) st
x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) )
in E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) holds
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
# x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
V57()
V58()
V59() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) is
convergent ) implies (
I : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
V57()
V58()
V59() )
Real_Sequence) is
convergent &
lim I : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
V57()
V58()
V59() )
Real_Sequence) : ( ( ) (
complex real ext-real )
Element of
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
= Integral (
M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ,
(lim F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like V58() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) (
V58() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
ext-real )
Element of
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) ) ) ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
S being ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
X : ( ( non
empty ) ( non
empty )
set ) )
for
M being ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
S : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
for
E being ( ( ) ( )
Element of
S : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) ) st
M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
. E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) (
ext-real )
Element of
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
< +infty : ( ( ) ( non
empty non
real ext-real positive non
negative )
set ) &
E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
= dom (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V40() V41() V42() V43() V44() V45() V46() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) & ( for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
. n : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58() )
ExtREAL_sequence) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) )
is_measurable_on E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ) &
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) is
uniformly_bounded & ( for
x being ( ( ) ( )
Element of
X : ( ( non
empty ) ( non
empty )
set ) ) st
x : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58() )
ExtREAL_sequence)
in E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) holds
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
# x : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58() )
ExtREAL_sequence) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
V57()
V58()
V59() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) is
convergent ) holds
( ( for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
. n : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58() )
ExtREAL_sequence) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) )
is_integrable_on M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ) &
lim F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like V58() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) (
V58() )
set ) ) : ( ( ) ( )
set ) )
is_integrable_on M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) & ex
I being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58() )
ExtREAL_sequence) st
( ( for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
I : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58() )
ExtREAL_sequence)
. n : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) : ( ( ) (
ext-real )
Element of
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
= Integral (
M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ,
(F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real ) Nat) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
ext-real )
Element of
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) ) &
I : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58() )
ExtREAL_sequence) is
convergent &
lim I : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58() )
ExtREAL_sequence) : ( ( ) (
ext-real )
Element of
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
= Integral (
M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ,
(lim F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like V58() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) (
V58() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
ext-real )
Element of
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) ) ) ;
definition
let X be ( ( ) ( )
set ) ;
let F be ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
X : ( ( ) ( )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
X : ( ( ) ( )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
X : ( ( ) ( )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( ) ( )
set ) ) ;
let f be ( (
Function-like ) (
Relation-like X : ( ( ) ( )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
PartFunc of ,) ;
pred F is_uniformly_convergent_to f means
(
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) is
with_the_same_dom &
dom (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) V57() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( V57() ) set ) ) : ( ( ) ( ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V40() V41() V42() V43() V44() V45() V46() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
complex Relation-like X : ( (
V41()
right_end ) (
V41()
right_end )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V41() )
Element of
K6(
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ) : ( ( ) ( )
set ) )
= dom f : ( (
Function-like V51(
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
Element of
K6(
K7(
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) (
V58() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V41() )
Element of
K6(
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ) : ( ( ) ( )
set ) ) & ( for
e being ( (
real ) (
complex real ext-real )
number ) st
e : ( (
real ) (
complex real ext-real )
number )
> 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non
positive non
negative V40()
V41()
V42()
V43()
V44()
V45()
V46()
V72()
V74() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ) holds
ex
N being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) st
for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat)
for
x being ( ( ) ( )
Element of
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ) st
n : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat)
>= N : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) &
x : ( ( ) ( )
Element of
X : ( ( ) ( )
set ) )
in dom (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) V57() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( V57() ) set ) ) : ( ( ) ( ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V40() V41() V42() V43() V44() V45() V46() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
complex Relation-like X : ( (
V41()
right_end ) (
V41()
right_end )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V41() )
Element of
K6(
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ) : ( ( ) ( )
set ) ) holds
|.(((F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) V57() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( V57() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real ) Nat) ) : ( ( Function-like ) ( complex Relation-like X : ( ( V41() right_end ) ( V41() right_end ) set ) -defined REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) -valued Function-like V57() V58() V59() ) Element of K6(K7(X : ( ( V41() right_end ) ( V41() right_end ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( V57() V58() V59() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) - (f : ( ( Function-like V51(F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) V57() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( V57() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V41() ) set ) ) V66() nonnegative sigma-additive ) ( non empty Relation-like F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) V57() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( V57() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V41() ) set ) -valued Function-like total V51(F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) V57() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( V57() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V41() ) set ) ) V58() V66() nonnegative sigma-additive ) Element of K6(K7(F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) V57() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( V57() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V41() ) set ) ) : ( ( ) ( V58() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) .| : ( ( ) (
complex real ext-real )
Element of
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
< e : ( (
real ) (
complex real ext-real )
number ) ) );
end;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
S being ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
X : ( ( non
empty ) ( non
empty )
set ) )
for
M being ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
S : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
for
E being ( ( ) ( )
Element of
S : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) )
for
f being ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
PartFunc of ,) st
M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
. E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) (
ext-real )
Element of
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
< +infty : ( ( ) ( non
empty non
real ext-real positive non
negative )
set ) &
E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
= dom (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V40() V41() V42() V43() V44() V45() V46() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) & ( for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
. n : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58() )
ExtREAL_sequence) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) )
is_integrable_on M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ) &
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
is_uniformly_convergent_to f : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
PartFunc of ,) holds
(
f : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
PartFunc of ,)
is_integrable_on M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) & ex
I being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58() )
ExtREAL_sequence) st
( ( for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
I : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58() )
ExtREAL_sequence)
. n : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) : ( ( ) (
ext-real )
Element of
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
= Integral (
M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ,
(F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real ) Nat) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
ext-real )
Element of
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) ) &
I : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58() )
ExtREAL_sequence) is
convergent &
lim I : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58() )
ExtREAL_sequence) : ( ( ) (
ext-real )
Element of
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
= Integral (
M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ,
f : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
PartFunc of ,) ) : ( ( ) (
ext-real )
Element of
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) ) ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
S being ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
X : ( ( non
empty ) ( non
empty )
set ) )
for
M being ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
S : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
for
E being ( ( ) ( )
Element of
S : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
for
P being ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
PartFunc of ,)
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) ) st
E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
= dom (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V40() V41() V42() V43() V44() V45() V46() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) &
E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
= dom P : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
PartFunc of ,) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) & ( for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
. n : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
is_measurable_on E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ) &
P : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
PartFunc of ,)
is_integrable_on M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) & ( for
x being ( ( ) ( )
Element of
X : ( ( non
empty ) ( non
empty )
set ) )
for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) st
x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) )
in E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) holds
|.(F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real ) Nat) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like V57() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( V57() ) set ) ) : ( ( ) ( ) set ) ) .| : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) (
complex real ext-real )
Element of
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
<= P : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
PartFunc of ,)
. x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) (
complex real ext-real )
Element of
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) ) & ( for
x being ( ( ) ( )
Element of
X : ( ( non
empty ) ( non
empty )
set ) ) st
x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) )
in E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) holds
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
# x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) is
convergent ) holds
lim F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
is_integrable_on M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
S being ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
X : ( ( non
empty ) ( non
empty )
set ) )
for
M being ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
S : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
for
E being ( ( ) ( )
Element of
S : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
for
P being ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
PartFunc of ,)
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) ) st
E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
= dom (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V40() V41() V42() V43() V44() V45() V46() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) &
E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
= dom P : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
PartFunc of ,) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) & ( for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
. n : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Complex_Sequence) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
is_measurable_on E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ) &
P : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
PartFunc of ,)
is_integrable_on M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) & ( for
x being ( ( ) ( )
Element of
X : ( ( non
empty ) ( non
empty )
set ) )
for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) st
x : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Complex_Sequence)
in E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) holds
|.(F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like V57() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( V57() ) set ) ) : ( ( ) ( ) set ) ) .| : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) (
V57()
V58()
V59() )
set ) ) : ( ( ) ( )
set ) )
. x : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Complex_Sequence) : ( ( ) (
complex real ext-real )
Element of
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
<= P : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set )
-valued Function-like V57()
V58()
V59() )
PartFunc of ,)
. x : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Complex_Sequence) : ( ( ) (
complex real ext-real )
Element of
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) ) holds
ex
I being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Complex_Sequence) st
( ( for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
I : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Complex_Sequence)
. n : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
= Integral (
M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ,
(F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
complex ) (
complex )
set ) ) & ( ( for
x being ( ( ) ( )
Element of
X : ( ( non
empty ) ( non
empty )
set ) ) st
x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) )
in E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) holds
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
# x : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) is
convergent ) implies (
I : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Complex_Sequence) is
convergent &
lim I : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Complex_Sequence) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
= Integral (
M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ,
(lim F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
complex ) (
complex )
set ) ) ) ) ;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
S being ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
X : ( ( non
empty ) ( non
empty )
set ) )
for
M being ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
S : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
for
E being ( ( ) ( )
Element of
S : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) ) st
M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
. E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) (
ext-real )
Element of
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
< +infty : ( ( ) ( non
empty non
real ext-real positive non
negative )
set ) &
E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
= dom (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V40() V41() V42() V43() V44() V45() V46() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) & ( for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
. n : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Complex_Sequence) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
is_measurable_on E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ) &
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) is
uniformly_bounded & ( for
x being ( ( ) ( )
Element of
X : ( ( non
empty ) ( non
empty )
set ) ) st
x : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Complex_Sequence)
in E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) holds
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
# x : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Complex_Sequence) : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) is
convergent ) holds
( ( for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
. n : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Complex_Sequence) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
is_integrable_on M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ) &
lim F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
is_integrable_on M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) & ex
I being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Complex_Sequence) st
( ( for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
I : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Complex_Sequence)
. n : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
= Integral (
M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ,
(F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real ) Nat) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
complex ) (
complex )
set ) ) &
I : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Complex_Sequence) is
convergent &
lim I : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Complex_Sequence) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
= Integral (
M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ,
(lim F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
complex ) (
complex )
set ) ) ) ;
definition
let X be ( ( ) ( )
set ) ;
let F be ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
X : ( ( ) ( )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
X : ( ( ) ( )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
X : ( ( ) ( )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) ) )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( ) ( )
set ) ) ;
let f be ( (
Function-like ) (
Relation-like X : ( ( ) ( )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
PartFunc of ,) ;
pred F is_uniformly_convergent_to f means
(
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) is
with_the_same_dom &
dom (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) V57() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( V57() ) set ) ) : ( ( ) ( ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V40() V41() V42() V43() V44() V45() V46() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
complex Relation-like X : ( (
V41()
right_end ) (
V41()
right_end )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V41() )
Element of
K6(
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ) : ( ( ) ( )
set ) )
= dom f : ( (
Function-like V51(
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
Element of
K6(
K7(
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) ) : ( ( ) (
V58() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V41() )
Element of
K6(
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ) : ( ( ) ( )
set ) ) & ( for
e being ( (
real ) (
complex real ext-real )
number ) st
e : ( (
real ) (
complex real ext-real )
number )
> 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non
positive non
negative V40()
V41()
V42()
V43()
V44()
V45()
V46()
V72()
V74() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ) holds
ex
N being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) st
for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat)
for
x being ( ( ) ( )
Element of
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ) st
n : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat)
>= N : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) &
x : ( ( ) ( )
Element of
X : ( ( ) ( )
set ) )
in dom (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) V57() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( V57() ) set ) ) : ( ( ) ( ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V40() V41() V42() V43() V44() V45() V46() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
complex Relation-like X : ( (
V41()
right_end ) (
V41()
right_end )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V41() )
Element of
K6(
X : ( (
V41()
right_end ) (
V41()
right_end )
set ) ) : ( ( ) ( )
set ) ) holds
|.(((F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) V57() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( V57() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real ) Nat) ) : ( ( Function-like ) ( complex Relation-like X : ( ( V41() right_end ) ( V41() right_end ) set ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like V57() ) Element of K6(K7(X : ( ( V41() right_end ) ( V41() right_end ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( V57() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) ) : ( ( ) ( complex ) set ) - (f : ( ( Function-like V51(F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) V57() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( V57() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V41() ) set ) ) V66() nonnegative sigma-additive ) ( non empty Relation-like F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) V57() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( V57() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V41() ) set ) -valued Function-like total V51(F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) V57() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( V57() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V41() ) set ) ) V58() V66() nonnegative sigma-additive ) Element of K6(K7(F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) V57() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( V57() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V41() ) set ) ) : ( ( ) ( V58() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) set ) ) : ( ( ) ( ) set ) .| : ( ( ) (
complex real ext-real )
Element of
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) )
< e : ( (
real ) (
complex real ext-real )
number ) ) );
end;
theorem
for
X being ( ( non
empty ) ( non
empty )
set )
for
S being ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
X : ( ( non
empty ) ( non
empty )
set ) )
for
M being ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
S : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
for
E being ( ( ) ( )
Element of
S : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
for
F being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
X : ( ( non
empty ) ( non
empty )
set ) )
for
f being ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
PartFunc of ,) st
M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
. E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) (
ext-real )
Element of
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
< +infty : ( ( ) ( non
empty non
real ext-real positive non
negative )
set ) &
E : ( ( ) ( )
Element of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
= dom (F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V40() V41() V42() V43() V44() V45() V46() V72() V74() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K6(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) & ( for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
. n : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Complex_Sequence) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) )
is_integrable_on M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ) &
F : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) ( non
empty functional )
set ) )
with_the_same_dom )
Functional_Sequence of ( ( ) ( non
empty functional )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
is_uniformly_convergent_to f : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
PartFunc of ,) holds
(
f : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
PartFunc of ,)
is_integrable_on M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) & ex
I being ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Complex_Sequence) st
( ( for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) holds
I : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Complex_Sequence)
. n : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural complex real ext-real )
Nat) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
= Integral (
M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ,
(F : ( ( Function-like V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V51( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V40() V41() V42() V43() V44() V45() V46() ) Element of K6(REAL : ( ( ) ( non empty V29() V40() V41() V42() V46() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V29() V40() V46() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real ) Nat) ) : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
Element of
K6(
K7(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) : ( ( ) (
V57() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
complex ) (
complex )
set ) ) &
I : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Complex_Sequence) is
convergent &
lim I : ( (
Function-like V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) ) ) ( non
empty Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like total V51(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V40()
V41()
V42()
V43()
V44()
V45()
V46() )
Element of
K6(
REAL : ( ( ) ( non
empty V29()
V40()
V41()
V42()
V46() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
V57() )
Complex_Sequence) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set ) )
= Integral (
M : ( (
Function-like V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V66()
nonnegative sigma-additive ) ( non
empty Relation-like b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) )
-defined ExtREAL : ( ( ) ( non
empty V41() )
set )
-valued Function-like total V51(
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ,
ExtREAL : ( ( ) ( non
empty V41() )
set ) )
V58()
V66()
nonnegative sigma-additive )
sigma_Measure of
b2 : ( ( non
empty compl-closed sigma-multiplicative ) ( non
empty compl-closed sigma-multiplicative )
SigmaField of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ,
f : ( (
Function-like ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V29()
V40()
V46() )
set )
-valued Function-like V57() )
PartFunc of ,) ) : ( (
complex ) (
complex )
set ) ) ) ;