:: NDIFF_4 semantic presentation

begin

theorem :: NDIFF_4:1
for m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for f1, f2 being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) holds f1 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) - f2 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ,(REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = f1 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) + (- f2 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ,(REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ,(REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

definition
let n be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) ;
let x be ( ( real ) ( V11() real ext-real ) number ) ;
pred f is_differentiable_in x means :: NDIFF_4:def 1
 ex g being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS n : ( ( ) ( ) NORMSTR ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st
( f : ( ( ) ( ) Element of n : ( ( ) ( ) NORMSTR ) ) = g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) & g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) is_differentiable_in x : ( ( Function-like quasi_total ) ( Relation-like K7(n : ( ( ) ( ) NORMSTR ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(n : ( ( ) ( ) NORMSTR ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) );
end;

theorem :: NDIFF_4:2
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,)
for h being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,)
for x being ( ( real ) ( V11() real ext-real ) number ) st h : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) is_differentiable_in x : ( ( real ) ( V11() real ext-real ) number ) iff h : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) is_differentiable_in x : ( ( real ) ( V11() real ext-real ) number ) ) ;

definition
let n be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) ;
let x be ( ( real ) ( V11() real ext-real ) number ) ;
func diff (f,x) -> ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(n : ( ( ) ( ) NORMSTR ) ) FinSequence-like ) Element of REAL n : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) means :: NDIFF_4:def 2
 ex g being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS n : ( ( ) ( ) NORMSTR ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st
( f : ( ( ) ( ) Element of n : ( ( ) ( ) NORMSTR ) ) = g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) & it : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = diff (g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) ,x : ( ( Function-like quasi_total ) ( Relation-like K7(n : ( ( ) ( ) NORMSTR ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(n : ( ( ) ( ) NORMSTR ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS n : ( ( ) ( ) NORMSTR ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) ) );
end;

theorem :: NDIFF_4:3
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,)
for h being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,)
for x being ( ( real ) ( V11() real ext-real ) number ) st h : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) holds
diff (f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) ,x : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) = diff (h : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) ,x : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) ;

definition
let n be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) ;
let X be ( ( ) ( ) set ) ;
pred f is_differentiable_on X means :: NDIFF_4:def 3
( X : ( ( Function-like quasi_total ) ( Relation-like K7(n : ( ( ) ( ) NORMSTR ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(n : ( ( ) ( ) NORMSTR ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) c= dom f : ( ( ) ( ) Element of n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( V160() V161() V162() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in X : ( ( Function-like quasi_total ) ( Relation-like K7(n : ( ( ) ( ) NORMSTR ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(n : ( ( ) ( ) NORMSTR ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( ) ( ) Element of n : ( ( ) ( ) NORMSTR ) ) | X : ( ( Function-like quasi_total ) ( Relation-like K7(n : ( ( ) ( ) NORMSTR ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(n : ( ( ) ( ) NORMSTR ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL n : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ,(REAL n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_in x : ( ( ) ( V11() real ext-real ) Real) ) );
end;

theorem :: NDIFF_4:4
for X being ( ( ) ( ) set )
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) is_differentiable_on X : ( ( ) ( ) set ) holds
X : ( ( ) ( ) set ) is ( ( ) ( V160() V161() V162() ) Subset of ( ( ) ( ) set ) ) ;

theorem :: NDIFF_4:5
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) iff ( Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) c= dom f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) : ( ( ) ( V160() V161() V162() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) holds
f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) is_differentiable_in x : ( ( ) ( V11() real ext-real ) Real) ) ) ) ;

theorem :: NDIFF_4:6
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for Y being ( ( ) ( V160() V161() V162() ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) is_differentiable_on Y : ( ( ) ( V160() V161() V162() ) Subset of ( ( ) ( ) set ) ) holds
Y : ( ( ) ( V160() V161() V162() ) Subset of ( ( ) ( ) set ) ) is open ;

definition
let n be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) ;
let X be ( ( ) ( ) set ) ;
assume f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) is_differentiable_on X : ( ( ) ( ) set ) ;
func f `| X -> ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL n : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) means :: NDIFF_4:def 4
( dom it : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like V160() V161() V162() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) = X : ( ( Function-like quasi_total ) ( Relation-like K7(n : ( ( ) ( ) NORMSTR ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(n : ( ( ) ( ) NORMSTR ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in X : ( ( Function-like quasi_total ) ( Relation-like K7(n : ( ( ) ( ) NORMSTR ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(n : ( ( ) ( ) NORMSTR ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
it : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( ) set ) = diff (f : ( ( ) ( ) Element of n : ( ( ) ( ) NORMSTR ) ) ,x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(n : ( ( ) ( ) NORMSTR ) ) FinSequence-like ) Element of REAL n : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) ) );
end;

theorem :: NDIFF_4:7
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) st Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) c= dom f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) : ( ( ) ( V160() V161() V162() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) & ex r being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) st rng f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) : ( ( ) ( functional V233() V234() V235() ) Element of K6((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) = {r : ( ( ) ( V11() real ext-real ) Real) } : ( ( ) ( non empty V160() V161() V162() ) set ) holds
( f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) holds
(f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) `| Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) /. x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( Relation-like Function-like complex-valued ext-real-valued real-valued V56(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) = 0* n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) ) ) ;

theorem :: NDIFF_4:8
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for x0 being ( ( real ) ( V11() real ext-real ) number )
for f being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,)
for g being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,)
for N being ( ( ) ( V160() V161() V162() open ) Neighbourhood of x0 : ( ( real ) ( V11() real ext-real ) number ) ) st f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) = g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) & f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) is_differentiable_in x0 : ( ( real ) ( V11() real ext-real ) number ) & N : ( ( ) ( V160() V161() V162() open ) Neighbourhood of b2 : ( ( real ) ( V11() real ext-real ) number ) ) c= dom f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) : ( ( ) ( V160() V161() V162() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) holds
for h being ( ( non-zero Function-like quasi_total 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V132() V159() V160() V161() V162() V163() V164() V165() V166() V233() V234() V235() V236() V237() V238() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( non empty Relation-like non-zero NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V132() V159() V160() V161() V162() V163() V164() V165() V166() V233() V234() V235() V236() V237() V238() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence)
for c being ( ( Function-like constant quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like constant total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) st rng c : ( ( Function-like constant quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like constant total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( trivial V160() V161() V162() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) = {x0 : ( ( real ) ( V11() real ext-real ) number ) } : ( ( ) ( non empty V160() V161() V162() ) set ) & rng (h : ( ( non-zero Function-like quasi_total 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V132() V159() V160() V161() V162() V163() V164() V165() V166() V233() V234() V235() V236() V237() V238() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( non empty Relation-like non-zero NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V132() V159() V160() V161() V162() V163() V164() V165() V166() V233() V234() V235() V236() V237() V238() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( Function-like constant quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like constant total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V160() V161() V162() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) c= N : ( ( ) ( V160() V161() V162() open ) Neighbourhood of b2 : ( ( real ) ( V11() real ext-real ) number ) ) holds
( (h : ( ( non-zero Function-like quasi_total 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V132() V159() V160() V161() V162() V163() V164() V165() V166() V233() V234() V235() V236() V237() V238() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( non empty Relation-like non-zero NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V132() V159() V160() V161() V162() V163() V164() V165() V166() V233() V234() V235() V236() V237() V238() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) ") : ( ( Function-like ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) ((g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) /* (h : ( ( non-zero Function-like quasi_total 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V132() V159() V160() V161() V162() V163() V164() V165() V166() V233() V234() V235() V236() V237() V238() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( non empty Relation-like non-zero NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V132() V159() V160() V161() V162() V163() V164() V165() V166() V233() V234() V235() V236() V237() V238() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( Function-like constant quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like constant total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) - (g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) /* c : ( ( Function-like constant quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like constant total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is convergent & diff (f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) ,x0 : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) = lim ((h : ( ( non-zero Function-like quasi_total 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V132() V159() V160() V161() V162() V163() V164() V165() V166() V233() V234() V235() V236() V237() V238() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( non empty Relation-like non-zero NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V132() V159() V160() V161() V162() V163() V164() V165() V166() V233() V234() V235() V236() V237() V238() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) ") : ( ( Function-like ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) ((g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) /* (h : ( ( non-zero Function-like quasi_total 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V132() V159() V160() V161() V162() V163() V164() V165() V166() V233() V234() V235() V236() V237() V238() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( non empty Relation-like non-zero NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V132() V159() V160() V161() V162() V163() V164() V165() V166() V233() V234() V235() V236() V237() V238() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( Function-like constant quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like constant total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) - (g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) /* c : ( ( Function-like constant quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like constant total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) ) ;

theorem :: NDIFF_4:9
for x0, r being ( ( ) ( V11() real ext-real ) Real)
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) is_differentiable_in x0 : ( ( ) ( V11() real ext-real ) Real) holds
( r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ,(REAL b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_in x0 : ( ( ) ( V11() real ext-real ) Real) & diff ((r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ,(REAL b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) = r : ( ( ) ( V11() real ext-real ) Real) * (diff (f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) ) ;

theorem :: NDIFF_4:10
for x0 being ( ( ) ( V11() real ext-real ) Real)
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) is_differentiable_in x0 : ( ( ) ( V11() real ext-real ) Real) holds
( - f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_in x0 : ( ( ) ( V11() real ext-real ) Real) & diff ((- f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) = - (diff (f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) ) ;

theorem :: NDIFF_4:11
for x0 being ( ( ) ( V11() real ext-real ) Real)
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for f1, f2 being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) st f1 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) is_differentiable_in x0 : ( ( ) ( V11() real ext-real ) Real) & f2 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) is_differentiable_in x0 : ( ( ) ( V11() real ext-real ) Real) holds
( f1 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) + f2 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_in x0 : ( ( ) ( V11() real ext-real ) Real) & diff ((f1 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) + f2 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) = (diff (f1 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) + (diff (f2 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) ) ;

theorem :: NDIFF_4:12
for x0 being ( ( ) ( V11() real ext-real ) Real)
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for f1, f2 being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) st f1 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) is_differentiable_in x0 : ( ( ) ( V11() real ext-real ) Real) & f2 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) is_differentiable_in x0 : ( ( ) ( V11() real ext-real ) Real) holds
( f1 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) - f2 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_in x0 : ( ( ) ( V11() real ext-real ) Real) & diff ((f1 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) - f2 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) = (diff (f1 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) - (diff (f2 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) ) ;

theorem :: NDIFF_4:13
for r being ( ( ) ( V11() real ext-real ) Real)
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) st Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) c= dom f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) : ( ( ) ( V160() V161() V162() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) holds
( r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) holds
((r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) set ) = r : ( ( ) ( V11() real ext-real ) Real) * (diff (f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) ,x : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) ) ) ;

theorem :: NDIFF_4:14
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) st Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) c= dom f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) : ( ( ) ( V160() V161() V162() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) holds
( - f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) holds
((- f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) set ) = - (diff (f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) ,x : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) ) ) ;

theorem :: NDIFF_4:15
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) )
for f1, f2 being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) st Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) c= dom (f1 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) + f2 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V160() V161() V162() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) & f1 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) & f2 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) holds
( f1 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) + f2 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) holds
((f1 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) + f2 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) set ) = (diff (f1 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) ,x : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) + (diff (f2 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) ,x : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) ) ) ;

theorem :: NDIFF_4:16
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) )
for f1, f2 being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) st Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) c= dom (f1 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) - f2 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V160() V161() V162() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) & f1 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) & f2 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) holds
( f1 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) - f2 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) holds
((f1 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) - f2 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) set ) = (diff (f1 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) ,x : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) - (diff (f2 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) ,x : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) ) ) ;

theorem :: NDIFF_4:17
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) st Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) c= dom f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) : ( ( ) ( V160() V161() V162() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) | Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is constant holds
( f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) holds
(f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) `| Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) set ) = 0* n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) ) ) ;

theorem :: NDIFF_4:18
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,)
for r, p being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) st Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) c= dom f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) : ( ( ) ( V160() V161() V162() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) holds
f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) /. x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( Relation-like Function-like complex-valued ext-real-valued real-valued V56(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) = (x : ( ( ) ( V11() real ext-real ) Real) * r : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) + p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) ) holds
( f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) holds
(f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) `| Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) set ) = r : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) ) ) ;

theorem :: NDIFF_4:19
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,)
for x0 being ( ( ) ( V11() real ext-real ) Real) st f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) is_differentiable_in x0 : ( ( ) ( V11() real ext-real ) Real) holds
f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: NDIFF_4:20
for X being ( ( ) ( ) set )
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) is_differentiable_on X : ( ( ) ( ) set ) holds
f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) | X : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is continuous ;

theorem :: NDIFF_4:21
for X being ( ( ) ( ) set )
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) is_differentiable_on X : ( ( ) ( ) set ) & Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) c= X : ( ( ) ( ) set ) holds
f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) ;

definition
let n be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) ;
attr f is differentiable means :: NDIFF_4:def 5
f : ( ( ) ( ) Element of n : ( ( ) ( ) NORMSTR ) ) is_differentiable_on dom f : ( ( ) ( ) Element of n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( V160() V161() V162() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ;
end;

registration
let n be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ;
cluster REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) --> (0* n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) -> Function-like quasi_total differentiable for ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like total quasi_total V239() V240() V241() ) Function of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) ;
end;

registration
let n be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ;
cluster non empty Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like total quasi_total V239() V240() V241() differentiable for ( ( ) ( ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ,(REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;
end;

theorem :: NDIFF_4:22
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like differentiable ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() differentiable ) PartFunc of ,) st Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) c= dom f : ( ( Function-like differentiable ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() differentiable ) PartFunc of ,) : ( ( ) ( V160() V161() V162() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Function-like differentiable ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() differentiable ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( V160() V161() V162() open ) Subset of ( ( ) ( ) set ) ) ;

theorem :: NDIFF_4:23
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for R being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) st R : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) is total holds
( R : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) is RestFunc-like iff for r being ( ( ) ( V11() real ext-real ) Real) st r : ( ( ) ( V11() real ext-real ) Real) > 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V132() V159() V160() V161() V162() V163() V164() V165() V166() V233() V234() V235() V236() V237() V238() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) holds
ex d being ( ( ) ( V11() real ext-real ) Real) st
( d : ( ( ) ( V11() real ext-real ) Real) > 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V132() V159() V160() V161() V162() V163() V164() V165() V166() V233() V234() V235() V236() V237() V238() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for z being ( ( ) ( V11() real ext-real ) Real) st z : ( ( ) ( V11() real ext-real ) Real) <> 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V132() V159() V160() V161() V162() V163() V164() V165() V166() V233() V234() V235() V236() V237() V238() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) & |.z : ( ( ) ( V11() real ext-real ) Real) .| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) < d : ( ( ) ( V11() real ext-real ) Real) holds
(|.z : ( ( ) ( V11() real ext-real ) Real) .| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ") : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) * ||.(R : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) /. z : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) .|| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) < r : ( ( ) ( V11() real ext-real ) Real) ) ) ) ;

theorem :: NDIFF_4:24
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for g being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( real ) ( V11() real ext-real ) number ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) & g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) is_differentiable_in x0 : ( ( real ) ( V11() real ext-real ) number ) holds
( (Proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) * g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_in x0 : ( ( real ) ( V11() real ext-real ) number ) & (Proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . (diff (g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) ,x0 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( ) ( ) Element of the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) = diff (((Proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) * g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,x0 : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) ) ;

theorem :: NDIFF_4:25
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for g being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( real ) ( V11() real ext-real ) number ) holds
( g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) is_differentiable_in x0 : ( ( real ) ( V11() real ext-real ) number ) iff for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(Proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) * g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_in x0 : ( ( real ) ( V11() real ext-real ) number ) ) ;

theorem :: NDIFF_4:26
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,)
for x0 being ( ( real ) ( V11() real ext-real ) number ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) & f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) is_differentiable_in x0 : ( ( real ) ( V11() real ext-real ) number ) holds
( (Proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_in x0 : ( ( real ) ( V11() real ext-real ) number ) & (Proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . (diff (f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) ,x0 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) = diff (((Proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,x0 : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) ) ;

theorem :: NDIFF_4:27
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,)
for x0 being ( ( real ) ( V11() real ext-real ) number ) holds
( f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) is_differentiable_in x0 : ( ( real ) ( V11() real ext-real ) number ) iff for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(Proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_in x0 : ( ( real ) ( V11() real ext-real ) number ) ) ;

theorem :: NDIFF_4:28
for X being ( ( ) ( ) set )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for g being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) & g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) is_differentiable_on X : ( ( ) ( ) set ) holds
( (Proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) * g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on X : ( ( ) ( ) set ) & (Proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) * (g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) `| X : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , the carrier of (REAL-NS b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = ((Proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) * g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) `| X : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: NDIFF_4:29
for X being ( ( ) ( ) set )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) & f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) is_differentiable_on X : ( ( ) ( ) set ) holds
( (Proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on X : ( ( ) ( ) set ) & (Proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) `| X : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = ((Proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) `| X : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: NDIFF_4:30
for X being ( ( ) ( ) set )
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for g being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) holds
( g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) is_differentiable_on X : ( ( ) ( ) set ) iff for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(Proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) * g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on X : ( ( ) ( ) set ) ) ;

theorem :: NDIFF_4:31
for X being ( ( ) ( ) set )
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) is_differentiable_on X : ( ( ) ( ) set ) iff for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(Proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -valued Function-like V239() V240() V241() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on X : ( ( ) ( ) set ) ) ;

theorem :: NDIFF_4:32
for J being ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty non trivial ) set ) , REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) )
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) ) st J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty non trivial ) set ) , REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) = proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like one-to-one total quasi_total onto bijective complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) holds
J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty non trivial ) set ) , REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) ) ;

theorem :: NDIFF_4:33
for x0 being ( ( ) ( V11() real ext-real ) Real)
for I being ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , ( ( ) ( non empty non trivial ) set ) ) st I : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , ( ( ) ( non empty non trivial ) set ) ) = (proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like one-to-one total quasi_total onto bijective complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) " : ( ( Relation-like Function-like ) ( Relation-like Function-like one-to-one ) set ) holds
I : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , ( ( ) ( non empty non trivial ) set ) ) is_continuous_in x0 : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: NDIFF_4:34
for S, T being ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for f1 being ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,)
for f2 being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in dom (f2 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) * f1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & f1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & f2 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in f1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) holds
f2 : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) * f1 : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: NDIFF_4:35
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty non trivial ) set ) , REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) )
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) )
for y0 being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) )
for g being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,)
for f being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) st J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty non trivial ) set ) , REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) = proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like one-to-one total quasi_total onto bijective complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) & x0 : ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) & y0 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) in dom g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( V160() V161() V162() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) & x0 : ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) ) = <*y0 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like ) FinSequence of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) & f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) = g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) * J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty non trivial ) set ) , REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) ) iff g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in y0 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) ;

theorem :: NDIFF_4:36
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for I being ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , ( ( ) ( non empty non trivial ) set ) )
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) )
for y0 being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) )
for g being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,)
for f being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) st I : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , ( ( ) ( non empty non trivial ) set ) ) = (proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like one-to-one total quasi_total onto bijective complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) " : ( ( Relation-like Function-like ) ( Relation-like Function-like one-to-one ) set ) & x0 : ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) & y0 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) in dom g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( V160() V161() V162() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) & x0 : ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) ) = <*y0 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like ) FinSequence of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) & f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) * I : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , ( ( ) ( non empty non trivial ) set ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) ) iff g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in y0 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) ;

theorem :: NDIFF_4:37
for x0 being ( ( ) ( V11() real ext-real ) Real)
for I being ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , ( ( ) ( non empty non trivial ) set ) ) st I : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , ( ( ) ( non empty non trivial ) set ) ) = (proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like one-to-one total quasi_total onto bijective complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) " : ( ( Relation-like Function-like ) ( Relation-like Function-like one-to-one ) set ) holds
( I : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , ( ( ) ( non empty non trivial ) set ) ) is_differentiable_in x0 : ( ( ) ( V11() real ext-real ) Real) & diff (I : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , ( ( ) ( non empty non trivial ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( ) Element of the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) = <*1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like complex-valued ext-real-valued real-valued natural-valued FinSequence-like ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

definition
let n be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like the carrier of (REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ;
let x be ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) ) ;
pred f is_differentiable_in x means :: NDIFF_4:def 6
 ex g being ( ( Function-like ) ( Relation-like REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ex y being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) st
( f : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) = g : ( ( Function-like ) ( Relation-like REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) & x : ( ( Function-like quasi_total ) ( Relation-like K7(n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( RAT : ( ( ) ( non empty V49() V160() V161() V162() V163() V166() ) set ) -valued INT : ( ( ) ( non empty V49() V160() V161() V162() V163() V164() V166() ) set ) -valued complex-valued ext-real-valued real-valued natural-valued ) set ) -defined n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued natural-valued ) Element of K6(K7(K7(n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( RAT : ( ( ) ( non empty V49() V160() V161() V162() V163() V166() ) set ) -valued INT : ( ( ) ( non empty V49() V160() V161() V162() V163() V164() V166() ) set ) -valued complex-valued ext-real-valued real-valued natural-valued ) set ) ,n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( RAT : ( ( ) ( non empty V49() V160() V161() V162() V163() V166() ) set ) -valued INT : ( ( ) ( non empty V49() V160() V161() V162() V163() V164() V166() ) set ) -valued complex-valued ext-real-valued real-valued natural-valued ) set ) ) : ( ( ) ( ) set ) ) = y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) & g : ( ( Function-like ) ( Relation-like REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_differentiable_in y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) );
end;

definition
let n be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like the carrier of (REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ;
let x be ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) ) ;
func diff (f,x) -> ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty non trivial ) set ) , REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) means :: NDIFF_4:def 7
 ex g being ( ( Function-like ) ( Relation-like REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ex y being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) st
( f : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) = g : ( ( Function-like ) ( Relation-like REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) & x : ( ( Function-like quasi_total ) ( Relation-like K7(n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( RAT : ( ( ) ( non empty V49() V160() V161() V162() V163() V166() ) set ) -valued INT : ( ( ) ( non empty V49() V160() V161() V162() V163() V164() V166() ) set ) -valued complex-valued ext-real-valued real-valued natural-valued ) set ) -defined n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued natural-valued ) Element of K6(K7(K7(n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( RAT : ( ( ) ( non empty V49() V160() V161() V162() V163() V166() ) set ) -valued INT : ( ( ) ( non empty V49() V160() V161() V162() V163() V164() V166() ) set ) -valued complex-valued ext-real-valued real-valued natural-valued ) set ) ,n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( RAT : ( ( ) ( non empty V49() V160() V161() V162() V163() V166() ) set ) -valued INT : ( ( ) ( non empty V49() V160() V161() V162() V163() V164() V166() ) set ) -valued complex-valued ext-real-valued real-valued natural-valued ) set ) ) : ( ( ) ( ) set ) ) = y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) & it : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ,n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( RAT : ( ( ) ( non empty V49() V160() V161() V162() V163() V166() ) set ) -valued INT : ( ( ) ( non empty V49() V160() V161() V162() V163() V164() V166() ) set ) -valued complex-valued ext-real-valued real-valued natural-valued ) set ) -defined n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued natural-valued ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ,n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( RAT : ( ( ) ( non empty V49() V160() V161() V162() V163() V166() ) set ) -valued INT : ( ( ) ( non empty V49() V160() V161() V162() V163() V164() V166() ) set ) -valued complex-valued ext-real-valued real-valued natural-valued ) set ) ,n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( RAT : ( ( ) ( non empty V49() V160() V161() V162() V163() V166() ) set ) -valued INT : ( ( ) ( non empty V49() V160() V161() V162() V163() V164() V166() ) set ) -valued complex-valued ext-real-valued real-valued natural-valued ) set ) ) : ( ( ) ( ) set ) ) = diff (g : ( ( Function-like ) ( Relation-like REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) );
end;

theorem :: NDIFF_4:38
for J being ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) , REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) )
for x0 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) st J : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) , REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) = proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like one-to-one total quasi_total onto bijective complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) holds
( J : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) , REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) is_differentiable_in x0 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) & diff (J : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) , REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ,x0 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued V56(1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) = J : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) , REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) ;

theorem :: NDIFF_4:39
for J being ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty non trivial ) set ) , REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) )
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) ) st J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty non trivial ) set ) , REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) = proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like one-to-one total quasi_total onto bijective complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) holds
( J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty non trivial ) set ) , REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) is_differentiable_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) ) & diff (J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty non trivial ) set ) , REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ,x0 : ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty non trivial ) set ) , REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) = J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty non trivial ) set ) , REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) ;

theorem :: NDIFF_4:40
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for I being ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , ( ( ) ( non empty non trivial ) set ) ) st I : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , ( ( ) ( non empty non trivial ) set ) ) = (proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like one-to-one total quasi_total onto bijective complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) " : ( ( Relation-like Function-like ) ( Relation-like Function-like one-to-one ) set ) holds
( ( for R being ( ( Function-like RestFunc-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like RestFunc-like ) RestFunc of REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) holds R : ( ( Function-like quasi_total V153( REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) V154( REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total V153( REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) V154( REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) ) LinearOperator of REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) * I : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , ( ( ) ( non empty non trivial ) set ) ) : ( ( Function-like ) ( non empty Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is ( ( Function-like RestFunc-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like RestFunc-like ) RestFunc of REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) ) & ( for L being ( ( Function-like quasi_total V153( REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) V154( REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total V153( REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) V154( REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) ) LinearOperator of REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) holds L : ( ( Function-like quasi_total V153( REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) V154( REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total V153( REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) V154( REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) ) LinearOperator of REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) * I : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , ( ( ) ( non empty non trivial ) set ) ) : ( ( Function-like ) ( non empty Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is ( ( Function-like quasi_total linear ) ( non empty Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total linear ) LinearFunc of REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) ) ) ;

theorem :: NDIFF_4:41
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty non trivial ) set ) , REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) st J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty non trivial ) set ) , REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) = proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like one-to-one total quasi_total onto bijective complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) holds
( ( for R being ( ( Function-like RestFunc-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like RestFunc-like ) RestFunc of REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) holds R : ( ( Function-like quasi_total linear ) ( non empty Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total linear ) LinearFunc of REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) * J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty non trivial ) set ) , REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( Function-like ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is ( ( Function-like RestFunc-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like RestFunc-like ) RestFunc of REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) ) & ( for L being ( ( Function-like quasi_total linear ) ( non empty Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total linear ) LinearFunc of REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) holds L : ( ( Function-like quasi_total linear ) ( non empty Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total linear ) LinearFunc of REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) * J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty non trivial ) set ) , REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( Function-like ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is ( ( Function-like quasi_total V153( REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) V154( REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) Lipschitzian ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total V153( REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) V154( REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) Lipschitzian ) LinearOperator of REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) ) ) ;

theorem :: NDIFF_4:42
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for I being ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , ( ( ) ( non empty non trivial ) set ) )
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) )
for y0 being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) )
for g being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,)
for f being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) st I : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , ( ( ) ( non empty non trivial ) set ) ) = (proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like one-to-one total quasi_total onto bijective complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) " : ( ( Relation-like Function-like ) ( Relation-like Function-like one-to-one ) set ) & x0 : ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) & y0 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) in dom g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( V160() V161() V162() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) & x0 : ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) ) = <*y0 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like ) FinSequence of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) & f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) * I : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , ( ( ) ( non empty non trivial ) set ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) & f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) is_differentiable_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) ) holds
( g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) is_differentiable_in y0 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) & diff (g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) ,y0 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) = (diff (f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) ,x0 : ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) ) )) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ,(REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) )) : ( ( non empty ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) . <*1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like complex-valued ext-real-valued real-valued natural-valued FinSequence-like ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) & ( for r being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) holds (diff (f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) ,x0 : ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) ) )) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ,(REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) )) : ( ( non empty ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) . <*r : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like ) FinSequence of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) = r : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) * (diff (g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) ,y0 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) )) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) ) ) ;

theorem :: NDIFF_4:43
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for I being ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , ( ( ) ( non empty non trivial ) set ) )
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) )
for y0 being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) )
for g being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,)
for f being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) st I : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , ( ( ) ( non empty non trivial ) set ) ) = (proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like one-to-one total quasi_total onto bijective complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) " : ( ( Relation-like Function-like ) ( Relation-like Function-like one-to-one ) set ) & x0 : ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) & y0 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) in dom g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( V160() V161() V162() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) & x0 : ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) ) = <*y0 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like ) FinSequence of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) & f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) * I : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , ( ( ) ( non empty non trivial ) set ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) is_differentiable_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) ) iff g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) is_differentiable_in y0 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) ;

theorem :: NDIFF_4:44
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty non trivial ) set ) , REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) )
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) )
for y0 being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) )
for g being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,)
for f being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) st J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty non trivial ) set ) , REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) = proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like one-to-one total quasi_total onto bijective complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) & x0 : ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) & y0 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) in dom g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( V160() V161() V162() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) & x0 : ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) ) = <*y0 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like ) FinSequence of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) & f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) = g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) * J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty non trivial ) set ) , REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) is_differentiable_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) ) iff g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) is_differentiable_in y0 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) ;

theorem :: NDIFF_4:45
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty non trivial ) set ) , REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) )
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) )
for y0 being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) )
for g being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,)
for f being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) st J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty non trivial ) set ) , REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) = proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like one-to-one total quasi_total onto bijective complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V233() V234() V235() ) FinSequenceSet of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) & x0 : ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) ) in dom f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) & y0 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) in dom g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( V160() V161() V162() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) & x0 : ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) ) = <*y0 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like ) FinSequence of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) & f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) = g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) * J : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of ( ( ) ( non empty non trivial ) set ) , REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) is_differentiable_in y0 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) is_differentiable_in x0 : ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) ) & diff (g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) ,y0 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) = (diff (f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) ,x0 : ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) ) )) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ,(REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) )) : ( ( non empty ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) . <*1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like complex-valued ext-real-valued real-valued natural-valued FinSequence-like ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) & ( for r being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) holds (diff (f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) ,x0 : ( ( ) ( ) Point of ( ( ) ( non empty non trivial ) set ) ) )) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ,(REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) )) : ( ( non empty ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) . <*r : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like ) FinSequence of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) = r : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) * (diff (g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) ,y0 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) )) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) ) ) ;

theorem :: NDIFF_4:46
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for R being ( ( Function-like RestFunc-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like RestFunc-like ) RestFunc of REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) st R : ( ( Function-like RestFunc-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like RestFunc-like ) RestFunc of REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) /. 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V132() V159() V160() V161() V162() V163() V164() V165() V166() V233() V234() V235() V236() V237() V238() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) = 0. (REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( V68( REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) holds
for e being ( ( ) ( V11() real ext-real ) Real) st e : ( ( ) ( V11() real ext-real ) Real) > 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V132() V159() V160() V161() V162() V163() V164() V165() V166() V233() V234() V235() V236() V237() V238() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) holds
ex d being ( ( ) ( V11() real ext-real ) Real) st
( d : ( ( ) ( V11() real ext-real ) Real) > 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V132() V159() V160() V161() V162() V163() V164() V165() V166() V233() V234() V235() V236() V237() V238() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for h being ( ( ) ( V11() real ext-real ) Real) st |.h : ( ( ) ( V11() real ext-real ) Real) .| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) < d : ( ( ) ( V11() real ext-real ) Real) holds
||.(R : ( ( Function-like RestFunc-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like RestFunc-like ) RestFunc of REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) /. h : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) .|| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) <= e : ( ( ) ( V11() real ext-real ) Real) * |.h : ( ( ) ( V11() real ext-real ) Real) .| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) ) ;

theorem :: NDIFF_4:47
for n, m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for R being ( ( Function-like RestFunc-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like RestFunc-like ) RestFunc of REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) )
for L being ( ( Function-like quasi_total V153( REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) V154( REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) Lipschitzian ) ( non empty Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total V153( REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) V154( REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) Lipschitzian ) LinearOperator of REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) holds L : ( ( Function-like quasi_total V153( REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) V154( REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) Lipschitzian ) ( non empty Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total V153( REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) V154( REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) Lipschitzian ) LinearOperator of REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) * R : ( ( Function-like RestFunc-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like RestFunc-like ) RestFunc of REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is ( ( Function-like RestFunc-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like RestFunc-like ) RestFunc of REAL-NS m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) ;

theorem :: NDIFF_4:48
for n, m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for R1 being ( ( Function-like RestFunc-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like RestFunc-like ) RestFunc of REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) st R1 : ( ( Function-like RestFunc-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like RestFunc-like ) RestFunc of REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) /. 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V132() V159() V160() V161() V162() V163() V164() V165() V166() V233() V234() V235() V236() V237() V238() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) = 0. (REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( V68( REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) holds
for R2 being ( ( Function-like RestFunc-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like RestFunc-like ) RestFunc of REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) st R2 : ( ( Function-like RestFunc-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like RestFunc-like ) RestFunc of REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) /. (0. (REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) : ( ( ) ( V68( REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) = 0. (REAL-NS m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( V68( REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) ) Element of the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) holds
for L being ( ( Function-like quasi_total linear ) ( non empty Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total linear ) LinearFunc of REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) holds R2 : ( ( Function-like RestFunc-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like RestFunc-like ) RestFunc of REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) * (L : ( ( Function-like quasi_total linear ) ( non empty Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total linear ) LinearFunc of REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) + R1 : ( ( Function-like RestFunc-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like RestFunc-like ) RestFunc of REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is ( ( Function-like RestFunc-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like RestFunc-like ) RestFunc of REAL-NS m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) ;

theorem :: NDIFF_4:49
for n, m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for R1 being ( ( Function-like RestFunc-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like RestFunc-like ) RestFunc of REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) st R1 : ( ( Function-like RestFunc-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like RestFunc-like ) RestFunc of REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) /. 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V132() V159() V160() V161() V162() V163() V164() V165() V166() V233() V234() V235() V236() V237() V238() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) = 0. (REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( V68( REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) holds
for R2 being ( ( Function-like RestFunc-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like RestFunc-like ) RestFunc of REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) st R2 : ( ( Function-like RestFunc-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like RestFunc-like ) RestFunc of REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) /. (0. (REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) : ( ( ) ( V68( REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) = 0. (REAL-NS m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( V68( REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) ) Element of the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) holds
for L1 being ( ( Function-like quasi_total linear ) ( non empty Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total linear ) LinearFunc of REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) )
for L2 being ( ( Function-like quasi_total V153( REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) V154( REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) Lipschitzian ) ( non empty Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total V153( REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) V154( REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) Lipschitzian ) LinearOperator of REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) holds (L2 : ( ( Function-like quasi_total V153( REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) V154( REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) Lipschitzian ) ( non empty Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total V153( REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) V154( REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) Lipschitzian ) LinearOperator of REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) * R1 : ( ( Function-like RestFunc-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like RestFunc-like ) RestFunc of REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) + (R2 : ( ( Function-like RestFunc-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like RestFunc-like ) RestFunc of REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) , REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) * (L1 : ( ( Function-like quasi_total linear ) ( non empty Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like total quasi_total linear ) LinearFunc of REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) + R1 : ( ( Function-like RestFunc-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like RestFunc-like ) RestFunc of REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is ( ( Function-like RestFunc-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like RestFunc-like ) RestFunc of REAL-NS m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ) ;

theorem :: NDIFF_4:50
for n, m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) )
for x0 being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) )
for g being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) st g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) is_differentiable_in x0 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) holds
for f being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) is_differentiable_in g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) /. x0 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) * g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_in x0 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) & diff ((f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) * g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) , the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) = (diff (f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) ,(g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) /. x0 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) )) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) ,(REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) )) : ( ( non empty ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) . (diff (g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) -valued Function-like ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) )) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V49() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() ) NORMSTR ) : ( ( ) ( non empty non trivial ) set ) ) ) ;