MESFUN7C

:: MESFUN7C semantic presentation

begin

definition

let X be ( ( non empty ) ( non empty ) set ) ;
let f be ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) ;

func R_EAL f -> ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( Relation-like Function-like FinSequence-like ) FinSequence of COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( Relation-like Function-like FinSequence-like ) FinSequence of COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( Relation-like Function-like FinSequence-like ) FinSequence of COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( ) ( Relation-like Function-like FinSequence-like ) FinSequence of COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) ) equals :: MESFUN7C:def 1
f : ( ( ) ( ) Element of K597(X : ( ( ) ( Relation-like Function-like FinSequence-like ) FinSequence of COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( ) M21( COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) )) ) ;

end;

theorem :: MESFUN7C:1

registration

let X be ( ( non empty ) ( non empty ) set ) ;
let f be ( ( Function-like V30(X : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) ) ( non empty Relation-like X : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like total V30(X : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) V45() V46() V47() ) Function of X : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) ;

cluster R_EAL f : ( ( Function-like V30(X : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) ) ( non empty Relation-like X : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like total V30(X : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) V45() V46() V47() ) Element of K6(K7(X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V39() ) set ) -valued Function-like V46() ) Element of K6(K7(X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty Relation-like V46() ) set ) ) : ( ( ) ( non empty ) set ) ) -> Function-like total ;

end;

definition

func inf f -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V39() ) set ) -valued Function-like V46() ) PartFunc of ,) equals :: MESFUN7C:def 2
inf (R_EAL f : ( ( ) ( ) set ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V39() ) set ) -valued Function-like V46() ) Element of K6(K7(X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( Relation-like V46() ) set ) ) : ( ( ) ( non empty ) set ) ) ;

end;

theorem :: MESFUN7C:2

definition

func sup f -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V39() ) set ) -valued Function-like V46() ) PartFunc of ,) equals :: MESFUN7C:def 3
sup (R_EAL f : ( ( ) ( ) set ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V39() ) set ) -valued Function-like V46() ) Element of K6(K7(X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( Relation-like V46() ) set ) ) : ( ( ) ( non empty ) set ) ) ;

end;

theorem :: MESFUN7C:3

definition

func inferior_realsequence f -> ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( ) ( ) set ) ) equals :: MESFUN7C:def 4
inferior_realsequence (R_EAL f : ( ( ) ( ) set ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( ) ( ) set ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty Relation-like ) set ) ) : ( ( ) ( non empty ) set ) ) ;

end;

theorem :: MESFUN7C:4

for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) 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 non negative ) Nat) holds
( dom ((inferior_realsequence f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) 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 non negative ) Nat) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V39() ) set ) -valued Function-like V46() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty Relation-like V46() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) = dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V38() V39() V40() V41() V42() V43() V60() V61() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( 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 dom ((inferior_realsequence f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) 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 non negative ) Nat) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V39() ) set ) -valued Function-like V46() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty Relation-like V46() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) holds
((inferior_realsequence f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) 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 non negative ) Nat) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V39() ) set ) -valued Function-like V46() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty Relation-like V46() ) set ) ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V39() ) set ) ) = (inferior_realsequence (R_EAL (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) 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 V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) V45() V46() V47() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V39() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V39() ) set ) ) V46() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty Relation-like V46() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V39() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V39() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V39() ) set ) ) V46() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty Relation-like V46() ) set ) ) : ( ( ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V39() ) set ) ) ) ) ;

definition

func superior_realsequence f -> ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( ) ( ) set ) ) equals :: MESFUN7C:def 5
superior_realsequence (R_EAL f : ( ( ) ( ) set ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( ) ( ) set ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty Relation-like ) set ) ) : ( ( ) ( non empty ) set ) ) ;

end;

theorem :: MESFUN7C:5

for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) 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 non negative ) Nat) holds
( dom ((superior_realsequence f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) 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 non negative ) Nat) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V39() ) set ) -valued Function-like V46() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty Relation-like V46() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) = dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V38() V39() V40() V41() V42() V43() V60() V61() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( 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 dom ((superior_realsequence f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) 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 non negative ) Nat) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V39() ) set ) -valued Function-like V46() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty Relation-like V46() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) holds
((superior_realsequence f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) 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 non negative ) Nat) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V39() ) set ) -valued Function-like V46() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty Relation-like V46() ) set ) ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V39() ) set ) ) = (superior_realsequence (R_EAL (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) 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 V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) V45() V46() V47() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V39() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V39() ) set ) ) V46() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty Relation-like V46() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V39() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V39() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V39() ) set ) ) V46() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty Relation-like V46() ) set ) ) : ( ( ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V39() ) set ) ) ) ) ;

theorem :: MESFUN7C:6

registration

let X be ( ( non empty ) ( non empty ) set ) ;
let f be ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) ;

cluster R_EAL f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty Relation-like ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) -> Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ;

end;

theorem :: MESFUN7C:7

theorem :: MESFUN7C:8

theorem :: MESFUN7C:9

theorem :: MESFUN7C:10

theorem :: MESFUN7C:11

definition

func lim_inf f -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V39() ) set ) -valued Function-like V46() ) PartFunc of ,) equals :: MESFUN7C:def 6
lim_inf (R_EAL f : ( ( ) ( ) set ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V39() ) set ) -valued Function-like V46() ) Element of K6(K7(X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( Relation-like V46() ) set ) ) : ( ( ) ( non empty ) set ) ) ;

end;

theorem :: MESFUN7C:12

definition

func lim_sup f -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V39() ) set ) -valued Function-like V46() ) PartFunc of ,) equals :: MESFUN7C:def 7
lim_sup (R_EAL f : ( ( ) ( ) set ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V39() ) set ) -valued Function-like V46() ) Element of K6(K7(X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( Relation-like V46() ) set ) ) : ( ( ) ( non empty ) set ) ) ;

end;

theorem :: MESFUN7C:13

definition

func lim f -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V39() ) set ) -valued Function-like V46() ) PartFunc of ,) equals :: MESFUN7C:def 8
lim (R_EAL f : ( ( ) ( ) set ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V39() ) set ) -valued Function-like V46() ) Element of K6(K7(X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( Relation-like V46() ) set ) ) : ( ( ) ( non empty ) set ) ) ;

end;

theorem :: MESFUN7C:14

theorem :: MESFUN7C:15

theorem :: MESFUN7C:16

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 V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) 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 ( ( b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of K6(K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of K6(K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of K6(K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) )
for r being ( ( real ) ( complex real ext-real ) number ) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) holds F : ( ( b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of K6(K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of K6(K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of K6(K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) : ( ( ) ( ) set ) = (dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V38() V39() V40() V41() V42() V43() V60() V61() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) /\ (great_dom ((f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) 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 non negative ) Nat) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( complex real ext-real ) number ) )) : ( ( ) ( ) set ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) holds
union (rng F : ( ( b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of K6(K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of K6(K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of K6(K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty ) Element of K6(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) = (dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V38() V39() V40() V41() V42() V43() V60() V61() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) /\ (great_dom ((sup f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) 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 V39() ) set ) -valued Function-like V46() ) PartFunc of ,) ,r : ( ( real ) ( complex real ext-real ) number ) )) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUN7C:17

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 V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) 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 ( ( b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of K6(K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of K6(K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of K6(K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) )
for r being ( ( real ) ( complex real ext-real ) number ) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) holds F : ( ( b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of K6(K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of K6(K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of K6(K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) : ( ( ) ( ) set ) = (dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V38() V39() V40() V41() V42() V43() V60() V61() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) /\ (great_eq_dom ((f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) 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 non negative ) Nat) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) ,r : ( ( real ) ( complex real ext-real ) number ) )) : ( ( ) ( ) set ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) holds
meet (rng F : ( ( b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of K6(K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of K6(K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of K6(K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty ) Element of K6(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) = (dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V38() V39() V40() V41() V42() V43() V60() V61() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) /\ (great_eq_dom ((inf f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) 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 V39() ) set ) -valued Function-like V46() ) PartFunc of ,) ,r : ( ( real ) ( complex real ext-real ) number ) )) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUN7C:18

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 V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,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 ) ) ) st dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V38() V39() V40() V41() V42() V43() V60() V61() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( 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 ) ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) holds f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) 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 non negative ) Nat) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) 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
lim_sup f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) 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 V39() ) set ) -valued Function-like V46() ) PartFunc of ,) 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 :: MESFUN7C:19

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 V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,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 ) ) ) st dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V38() V39() V40() V41() V42() V43() V60() V61() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( 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 ) ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) holds f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) 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 non negative ) Nat) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) 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
lim_inf f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) 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 V39() ) set ) -valued Function-like V46() ) PartFunc of ,) 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 :: MESFUN7C:20

theorem :: MESFUN7C:21

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 V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,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 ) ) ) st dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V38() V39() V40() V41() V42() V43() V60() V61() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( 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 ) ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) holds f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) 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 V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) 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 V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) 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 V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) V45() V46() V47() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) is convergent ) holds
lim f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) 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 V39() ) set ) -valued Function-like V46() ) PartFunc of ,) 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 :: MESFUN7C:22

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 V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for g being ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V39() ) set ) -valued Function-like V46() ) PartFunc of ,)
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 ) ) ) st dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V38() V39() V40() V41() V42() V43() V60() V61() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( 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 ) ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) holds f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) 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 V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) 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 ) ) ) ) & dom g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V39() ) set ) -valued Function-like V46() ) PartFunc of ,) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( 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 ) ) ) & ( 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 V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) 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 V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) V45() V46() V47() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) is convergent & g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V39() ) set ) -valued Function-like V46() ) PartFunc of ,) . x : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V39() ) set ) ) = lim (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) 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 V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) V45() V46() V47() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) ) ) holds
g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V39() ) set ) -valued Function-like V46() ) PartFunc of ,) 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 ) ) ) ;

begin

definition

let X be ( ( non empty ) ( non empty ) set ) ;
let H be ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) ;
let x be ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) ;

func H # x -> ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) V45() ) Complex_Sequence) means :: MESFUN7C:def 9
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) holds it : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(X : ( ( ) ( ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( Relation-like V45() ) set ) ) : ( ( ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) : ( ( ) ( complex ) set ) = (H : ( ( ) ( ) set ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(X : ( ( ) ( ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( Relation-like V45() ) set ) ) : ( ( ) ( non empty ) set ) ) . x : ( ( complex ) ( complex ) set ) : ( ( ) ( complex ) set ) ;

end;

definition

let X be ( ( non empty ) ( non empty ) set ) ;
let f be ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) ;

func lim f -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) means :: MESFUN7C:def 10
( dom it : ( ( complex ) ( complex ) set ) : ( ( ) ( ) Element of K6(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) = dom (f : ( ( ) ( ) set ) . 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V38() V39() V40() V41() V42() V43() V60() V61() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(X : ( ( ) ( ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( Relation-like V45() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) & ( for x being ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) st x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) in dom it : ( ( complex ) ( complex ) set ) : ( ( ) ( ) Element of K6(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) holds
it : ( ( complex ) ( complex ) set ) . x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) = lim (f : ( ( ) ( ) set ) # x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) V45() ) Complex_Sequence) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) ) );

end;

definition

let X be ( ( non empty ) ( non empty ) set ) ;
let f be ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) ;

func Re f -> ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( ) ( ) set ) ) means :: MESFUN7C:def 11
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) holds
( dom (it : ( ( complex ) ( complex ) set ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) = dom (f : ( ( ) ( ) set ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(X : ( ( ) ( ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( Relation-like V45() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) & ( for x being ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) st x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) in dom (it : ( ( complex ) ( complex ) set ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) holds
(it : ( ( complex ) ( complex ) set ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) = (Re (f : ( ( ) ( ) set ) # x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) V45() ) Complex_Sequence) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) V45() V46() V47() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) ) );

end;

registration

let X be ( ( non empty ) ( non empty ) set ) ;
let f be ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) ;

cluster Re f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (X : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty Relation-like ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) -> Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ;

end;

definition

let X be ( ( non empty ) ( non empty ) set ) ;
let f be ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) ;

func Im f -> ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( ) ( ) set ) ) means :: MESFUN7C:def 12
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) holds
( dom (it : ( ( complex ) ( complex ) set ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) = dom (f : ( ( ) ( ) set ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(X : ( ( ) ( ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( Relation-like V45() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) & ( for x being ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) st x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) in dom (it : ( ( complex ) ( complex ) set ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) holds
(it : ( ( complex ) ( complex ) set ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(X : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) = (Im (f : ( ( ) ( ) set ) # x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) V45() ) Complex_Sequence) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) V45() V46() V47() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) ) );

end;

registration

let X be ( ( non empty ) ( non empty ) set ) ;
let f be ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) ;

cluster Im f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (X : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty Relation-like ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) -> Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ;

end;

theorem :: MESFUN7C:23

for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( 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 dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V38() V39() V40() V41() V42() V43() V60() V61() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) holds
( (Re f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) 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 V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) V45() V46() V47() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) = Re (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) 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 V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) V45() ) Complex_Sequence) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) V45() V46() V47() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) & (Im f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) 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 V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) V45() V46() V47() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) = Im (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) 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 V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) V45() ) Complex_Sequence) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) V45() V46() V47() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MESFUN7C:24

theorem :: MESFUN7C:25

for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) 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 dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V38() V39() V40() V41() V42() V43() V60() V61() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) holds
f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) 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 V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) V45() ) Complex_Sequence) is convergent ) holds
( lim (Re f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) 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 V39() ) set ) -valued Function-like V46() ) PartFunc of ,) = Re (lim f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) 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 V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) & lim (Im f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) 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 V39() ) set ) -valued Function-like V46() ) PartFunc of ,) = Im (lim f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) 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 V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MESFUN7C:26

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 V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,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 ) ) ) st dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V38() V39() V40() V41() V42() V43() V60() V61() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( 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 ) ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) holds f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) 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 V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() ) set ) ) : ( ( ) ( non empty ) 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 V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) 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 V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) V45() ) Complex_Sequence) is convergent ) holds
lim f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) 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 V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) 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 :: MESFUN7C:27

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 V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for g being ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,)
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 ) ) ) st dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V38() V39() V40() V41() V42() V43() V60() V61() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( 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 ) ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) holds f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) 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 V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() ) set ) ) : ( ( ) ( non empty ) 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 ) ) ) ) & dom g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( 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 ) ) ) & ( 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 V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) 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 V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) V45() ) Complex_Sequence) is convergent & g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) . x : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( complex ) set ) = lim (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) 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 V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) , COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) V45() ) Complex_Sequence) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) ) ) holds
g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) 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 ) ) ) ;

begin

theorem :: MESFUN7C:28

for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,)
for r being ( ( ) ( complex real ext-real ) Real) holds (r : ( ( ) ( complex real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() ) set ) ) : ( ( ) ( non empty ) set ) ) | Y : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined b₂ : ( ( ) ( ) set ) -defined b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() ) set ) ) : ( ( ) ( non empty ) set ) ) = r : ( ( ) ( complex real ext-real ) Real) (#) (f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) | Y : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined b₂ : ( ( ) ( ) set ) -defined b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUN7C:29

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 ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,)
for k being ( ( real ) ( complex real ext-real ) number )
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 ) ) ) st 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V38() V39() V40() V41() V42() V43() V60() V61() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) <= k : ( ( real ) ( complex real ext-real ) number ) & 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 ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) 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 ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) .| : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) to_power k : ( ( real ) ( complex real ext-real ) number ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) 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 :: MESFUN7C:30

for X being ( ( non empty ) ( non empty ) set )
for f, g being ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) PartFunc of ,) holds (R_EAL f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V39() ) set ) -valued Function-like V46() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty Relation-like V46() ) set ) ) : ( ( ) ( non empty ) set ) ) (#) (R_EAL g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V39() ) set ) -valued Function-like V46() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty Relation-like V46() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V39() ) set ) -valued Function-like V46() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty Relation-like V46() ) set ) ) : ( ( ) ( non empty ) set ) ) = R_EAL (f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) PartFunc of ,) (#) g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V39() ) set ) -valued Function-like V46() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( non empty Relation-like V46() ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUN7C:31

theorem :: MESFUN7C:32

for X being ( ( non empty ) ( non empty ) set )
for f, g being ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) holds
( Re (f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) (#) g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) = ((Re f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) (#) (Re g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) - ((Im f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) (#) (Im g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) & Im (f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) (#) g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) = ((Im f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) (#) (Re g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) + ((Re f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) (#) (Im g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MESFUN7C:33

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, g being ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,)
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 ) ) ) st (dom f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) /\ (dom g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( 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 ) ) ) & f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) 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 ) ) ) & g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) 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 ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) (#) g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() ) set ) ) : ( ( ) ( non empty ) 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 :: MESFUN7C:34

theorem :: MESFUN7C:35

theorem :: MESFUN7C:36

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 V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V39() ) set ) ) V54() 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 V39() ) set ) -valued Function-like total V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V39() ) set ) ) V46() V54() 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 ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V39() ) set ) ) V54() 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 V39() ) set ) -valued Function-like total V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V39() ) set ) ) V46() V54() 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
ex F being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) st
( ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) holds F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) : ( ( ) ( ) set ) = (dom f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) /\ (great_eq_dom (|.f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) .| : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) ,(R_EAL (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V38() V39() V40() V41() V42() V43() V60() V61() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) / (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V38() V39() V40() V41() V42() V43() V60() V61() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V38() V39() V40() V41() V42() V43() V60() V61() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V39() ) set ) ) )) : ( ( ) ( ) set ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) & (dom f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) \ (eq_dom (|.f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) .| : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) ,0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V38() V39() V40() V41() V42() V43() V60() V61() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( ) ( ) set ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) = union (rng F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Element of K6(K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) holds
( F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) : ( ( ) ( ) set ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) & M : ( ( Function-like V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V39() ) set ) ) V54() 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 V39() ) set ) -valued Function-like total V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V39() ) set ) ) V46() V54() 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 V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ,b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) ) : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V39() ) set ) ) < +infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty V39() ) set ) ) ) ) ) ;

theorem :: MESFUN7C:37

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 V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,)
for A being ( ( ) ( ) set ) holds |.f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) .| : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) | A : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined b₃ : ( ( ) ( ) set ) -defined b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) = |.(f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) | A : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined b₃ : ( ( ) ( ) set ) -defined b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() ) set ) ) : ( ( ) ( non empty ) set ) ) .| : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUN7C:38

theorem :: MESFUN7C:39

for X being ( ( non empty ) ( non empty ) set )
for f, g being ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) holds (|.f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) .| : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) | (dom |.(f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() ) set ) ) : ( ( ) ( non empty ) set ) ) .| : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined dom |.(b₂ : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) + b₃ : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() ) set ) ) : ( ( ) ( non empty ) set ) ) .| : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) -defined b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) + (|.g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) .| : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) | (dom |.(f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() ) set ) ) : ( ( ) ( non empty ) set ) ) .| : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined dom |.(b₂ : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) + b₃ : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() ) set ) ) : ( ( ) ( non empty ) set ) ) .| : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) -defined b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) = (|.f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) .| : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) + |.g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) .| : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) | (dom |.(f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() ) set ) ) : ( ( ) ( non empty ) set ) ) .| : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined dom |.(b₂ : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) + b₃ : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() ) set ) ) : ( ( ) ( non empty ) set ) ) .| : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) -defined b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUN7C:40

theorem :: MESFUN7C:41

for X being ( ( non empty ) ( non empty ) set )
for f, g being ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) PartFunc of ,) st ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) PartFunc of ,) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) holds
f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) <= g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) ) holds
g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) PartFunc of ,) - f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) is nonnegative ;

theorem :: MESFUN7C:42

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 V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V39() ) set ) ) V54() 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 V39() ) set ) -valued Function-like total V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V39() ) set ) ) V46() V54() 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, g being ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V39() ) set ) ) V54() 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 V39() ) set ) -valued Function-like total V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V39() ) set ) ) V46() V54() 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 ) ) ) & g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V39() ) set ) ) V54() 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 V39() ) set ) -valued Function-like total V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V39() ) set ) ) V46() V54() 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
ex 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 ) ) ) 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 ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) & Integral (M : ( ( Function-like V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V39() ) set ) ) V54() 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 V39() ) set ) -valued Function-like total V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V39() ) set ) ) V46() V54() 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 V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() ) set ) ) : ( ( ) ( non empty ) set ) ) .| : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( 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 ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined b₆ : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) -defined b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V39() ) set ) ) <= (Integral (M : ( ( Function-like V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V39() ) set ) ) V54() 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 V39() ) set ) -valued Function-like total V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V39() ) set ) ) V46() V54() 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 V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) .| : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( 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 ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined b₆ : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) -defined b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V39() ) set ) ) + (Integral (M : ( ( Function-like V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V39() ) set ) ) V54() 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 V39() ) set ) -valued Function-like total V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V39() ) set ) ) V46() V54() 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 ) ) ) ,(|.g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) .| : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( 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 ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined b₆ : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) -defined b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V39() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V39() ) set ) ) ) ;

begin

definition

pred f is_simple_func_in S means :: MESFUN7C:def 13
ex F being ( ( V92() ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined S : ( ( ) ( ) set ) -valued Function-like FinSequence-like V92() ) Finite_Sep_Sequence of S : ( ( ) ( ) set ) ) st
( dom f : ( ( complex ) ( complex ) set ) : ( ( ) ( ) Element of K6(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) = union (rng F : ( ( V92() ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like V92() ) Finite_Sep_Sequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of K6(S : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat)
for x, y being ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) in dom F : ( ( V92() ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like V92() ) Finite_Sep_Sequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V38() V39() V40() V41() V42() V43() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ) & x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) in F : ( ( V92() ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like V92() ) Finite_Sep_Sequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) : ( ( ) ( ) set ) & y : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) in F : ( ( V92() ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like V92() ) Finite_Sep_Sequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) : ( ( ) ( ) set ) holds
f : ( ( complex ) ( complex ) set ) . x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) = f : ( ( complex ) ( complex ) set ) . y : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) );

end;

definition

let X be ( ( non empty ) ( non empty ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) PartFunc of ,) ;
let F be ( ( V92() ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like V92() ) Finite_Sep_Sequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let a be ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() FinSequence-like ) FinSequence of REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) ;

pred F,a are_Re-presentation_of f means :: MESFUN7C:def 14
( dom f : ( ( complex ) ( complex ) set ) : ( ( ) ( ) Element of K6(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) = union (rng F : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(X : ( ( ) ( ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( Relation-like V45() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V38() ) Element of K6(S : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) & dom F : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(X : ( ( ) ( ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( Relation-like V45() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V38() V39() V40() V41() V42() V43() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ) = dom a : ( ( ) ( ) Element of S : ( ( ) ( ) set ) ) : ( ( ) ( V38() V39() V40() V41() V42() V43() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) in dom F : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(X : ( ( ) ( ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( Relation-like V45() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V38() V39() V40() V41() V42() V43() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ) holds
for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in F : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(X : ( ( ) ( ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( Relation-like V45() ) set ) ) : ( ( ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) : ( ( ) ( complex ) set ) holds
f : ( ( complex ) ( complex ) set ) . x : ( ( ) ( ) set ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) = a : ( ( ) ( ) Element of S : ( ( ) ( ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) ) );

end;

definition

let X be ( ( non empty ) ( non empty ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) ;
let F be ( ( V92() ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like V92() ) Finite_Sep_Sequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let a be ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() FinSequence-like ) FinSequence of COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) ;

pred F,a are_Re-presentation_of f means :: MESFUN7C:def 15
( dom f : ( ( complex ) ( complex ) set ) : ( ( ) ( ) Element of K6(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) = union (rng F : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(X : ( ( ) ( ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( Relation-like V45() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V38() ) Element of K6(S : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) & dom F : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(X : ( ( ) ( ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( Relation-like V45() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V38() V39() V40() V41() V42() V43() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ) = dom a : ( ( ) ( ) Element of S : ( ( ) ( ) set ) ) : ( ( ) ( V38() V39() V40() V41() V42() V43() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) in dom F : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(X : ( ( ) ( ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( Relation-like V45() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V38() V39() V40() V41() V42() V43() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ) holds
for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in F : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) Element of K6(K7(X : ( ( ) ( ) set ) ,COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( Relation-like V45() ) set ) ) : ( ( ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) : ( ( ) ( complex ) set ) holds
f : ( ( complex ) ( complex ) set ) . x : ( ( ) ( ) set ) : ( ( ) ( ) set ) = a : ( ( ) ( ) Element of S : ( ( ) ( ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) : ( ( ) ( ) set ) ) );

end;

theorem :: MESFUN7C:43

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 ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) 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 ) ) iff ( Re f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) 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 ) ) & Im f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) 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 :: MESFUN7C:44

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 ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) 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
ex F being ( ( V92() ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like V92() ) Finite_Sep_Sequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ex a being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() FinSequence-like ) FinSequence of COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) st
( dom f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) = union (rng F : ( ( V92() ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like V92() ) Finite_Sep_Sequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of K6(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) & dom F : ( ( V92() ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like V92() ) Finite_Sep_Sequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V38() V39() V40() V41() V42() V43() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ) = dom a : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() FinSequence-like ) FinSequence of COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( V38() V39() V40() V41() V42() V43() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) in dom F : ( ( V92() ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like V92() ) Finite_Sep_Sequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V38() V39() V40() V41() V42() V43() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ) holds
for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in F : ( ( V92() ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like V92() ) Finite_Sep_Sequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) : ( ( ) ( ) set ) holds
f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( complex ) set ) = a : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() FinSequence-like ) FinSequence of COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) : ( ( ) ( complex ) set ) ) ) ;

theorem :: MESFUN7C:45

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 ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) 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 ) ) iff ex F being ( ( V92() ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like V92() ) Finite_Sep_Sequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ex a being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() FinSequence-like ) FinSequence of COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) st F : ( ( V92() ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like V92() ) Finite_Sep_Sequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ,a : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() FinSequence-like ) FinSequence of COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) are_Re-presentation_of f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) ) ;

theorem :: MESFUN7C:46

for c being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() FinSequence-like ) FinSequence of COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) )
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) in dom (Re c : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() FinSequence-like ) FinSequence of COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() FinSequence-like ) FinSequence of REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( V38() V39() V40() V41() V42() V43() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ) holds
(Re c : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() FinSequence-like ) FinSequence of COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() FinSequence-like ) FinSequence of REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) = Re (c : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() FinSequence-like ) FinSequence of COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) ) : ( ( ) ( complex ) set ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) ;

theorem :: MESFUN7C:47

for c being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() FinSequence-like ) FinSequence of COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) )
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) in dom (Im c : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() FinSequence-like ) FinSequence of COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() FinSequence-like ) FinSequence of REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( V38() V39() V40() V41() V42() V43() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ) holds
(Im c : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() FinSequence-like ) FinSequence of COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() FinSequence-like ) FinSequence of REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) = Im (c : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() FinSequence-like ) FinSequence of COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) ) : ( ( ) ( complex ) set ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) ;

theorem :: MESFUN7C:48

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 ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,)
for F being ( ( V92() ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like V92() ) Finite_Sep_Sequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) )
for a being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() FinSequence-like ) FinSequence of COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) holds
( F : ( ( V92() ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like V92() ) Finite_Sep_Sequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ,a : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() FinSequence-like ) FinSequence of COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) are_Re-presentation_of f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) iff ( F : ( ( V92() ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like V92() ) Finite_Sep_Sequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) , Re a : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() FinSequence-like ) FinSequence of COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() FinSequence-like ) FinSequence of REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) are_Re-presentation_of Re f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) & F : ( ( V92() ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like V92() ) Finite_Sep_Sequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) , Im a : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() FinSequence-like ) FinSequence of COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() FinSequence-like ) FinSequence of REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) are_Re-presentation_of Im f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: MESFUN7C:49

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 ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) 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 ) ) iff ex F being ( ( V92() ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like V92() ) Finite_Sep_Sequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ex c being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() FinSequence-like ) FinSequence of COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) st
( dom f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) : ( ( ) ( ) Element of K6(b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) = union (rng F : ( ( V92() ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like V92() ) Finite_Sep_Sequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of K6(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) & dom F : ( ( V92() ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like V92() ) Finite_Sep_Sequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V38() V39() V40() V41() V42() V43() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ) = dom c : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() FinSequence-like ) FinSequence of COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) : ( ( ) ( V38() V39() V40() V41() V42() V43() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) in dom F : ( ( V92() ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like V92() ) Finite_Sep_Sequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V38() V39() V40() V41() V42() V43() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ) holds
for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in F : ( ( V92() ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like V92() ) Finite_Sep_Sequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) : ( ( ) ( ) set ) holds
(Re f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) = (Re c : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() FinSequence-like ) FinSequence of COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() FinSequence-like ) FinSequence of REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) in dom F : ( ( V92() ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like V92() ) Finite_Sep_Sequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V38() V39() V40() V41() V42() V43() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ) holds
for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in F : ( ( V92() ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like FinSequence-like V92() ) Finite_Sep_Sequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) : ( ( ) ( ) set ) holds
(Im f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() ) Element of K6(K7(b₁ : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty Relation-like V45() V46() V47() ) set ) ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) = (Im c : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) -valued Function-like V45() FinSequence-like ) FinSequence of COMPLEX : ( ( ) ( non empty V33() V38() V44() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V38() V39() V40() V41() V42() V43() V44() ) Element of K6(REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) : ( ( ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) -valued Function-like V45() V46() V47() FinSequence-like ) FinSequence of REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Nat) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V33() V38() V39() V40() V44() ) set ) ) ) ) ) ;