:: PDIFF_9 semantic presentation

begin

theorem :: PDIFF_9:1
for S, T being ( ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) RealNormSpace)
for f being ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) )
for r being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional FinSequence-membered V160() V195() V196() V197() V198() V199() V200() V201() V202() V263() V264() V265() V266() V267() V268() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= r : ( ( ) ( V11() real ext-real ) Real) & ( for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st ||.x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) <= 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) holds
||.(f : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) <= r : ( ( ) ( V11() real ext-real ) Real) * ||.x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) holds
||.f : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) <= r : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: PDIFF_9:2
for Z being ( ( ) ( ) set )
for S being ( ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_continuous_on Z : ( ( ) ( ) set ) iff ( Z : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for s1 being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) st rng s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) c= Z : ( ( ) ( ) set ) & s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) is convergent & lim s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) in Z : ( ( ) ( ) set ) holds
( f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) /* s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like total quasi_total complex-yielding V186() V187() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is convergent & f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) /. (lim s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) = lim (f : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) /* s1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b2 : ( ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like total quasi_total complex-yielding V186() V187() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) ) ) ;

theorem :: PDIFF_9:3
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) holds dom (<>* f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Element of K6((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) = dom f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Element of K6((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) ;

theorem :: PDIFF_9:4
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) st Z : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Element of K6((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) holds
dom ((<>* f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Element of K6((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) = Z : ( ( ) ( ) set ) ;

theorem :: PDIFF_9:5
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) holds <>* (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) | Z : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = (<>* f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: PDIFF_9:6
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,)
for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in dom f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Element of K6((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) holds
( (<>* f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) set ) = <*(f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) . x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) *> : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) FinSequence of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) & (<>* f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like V33() V40(1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) = <*(f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) /. x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) *> : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) FinSequence of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ;

theorem :: PDIFF_9:7
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f, g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) holds
( <>* (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = (<>* f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) + (<>* g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & <>* (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = (<>* f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) - (<>* g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: PDIFF_9:8
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,)
for r being ( ( real ) ( V11() real ext-real ) number ) holds <>* (r : ( ( real ) ( V11() real ext-real ) number ) (#) f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = r : ( ( real ) ( V11() real ext-real ) number ) (#) (<>* f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: PDIFF_9:9
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,)
for g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) st <>* f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) holds
|.f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) .| : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) = |.g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) .| : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: PDIFF_9:10
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for Y being ( ( ) ( ) Subset of ) st X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) = Y : ( ( ) ( ) Subset of ) holds
( X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) is open iff Y : ( ( ) ( ) Subset of ) is open ) ;

theorem :: PDIFF_9:11
for i, j being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for q being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) holds
|.((reproj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ,(0* j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -defined REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like total quasi_total V269() V270() V271() ) Element of K6(K7(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ,(REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . q : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like V33() V40(b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) = |.q : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) .| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ;

theorem :: PDIFF_9:12
for j, i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) holds x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) = (reproj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -defined REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like total quasi_total V269() V270() V271() ) Element of K6(K7(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ,(REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . ((proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like total quasi_total complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like V33() V40(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ;

begin

theorem :: PDIFF_9:13
for m, n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) is_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) holds
X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) is open ;

theorem :: PDIFF_9:14
for m, n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) st X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) is open holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) is_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) iff ( X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) c= dom f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Element of K6((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) holds
f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) is_differentiable_in x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) ) ) ;

definition
let m, n be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let Z be ( ( ) ( ) set ) ;
let f be ( ( Function-like ) ( Relation-like REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) ;
assume Z : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Element of K6((REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) ;
func f `| Z -> ( ( Function-like ) ( Relation-like REAL m : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined Funcs ((REAL m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL n : ( ( ) ( ) Element of m : ( ( ) ( ) NORMSTR ) ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of REAL m : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) , REAL n : ( ( ) ( ) Element of m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) -valued Function-like ) PartFunc of ,) means :: PDIFF_9:def 1
( dom it : ( ( Function-like quasi_total ) ( Relation-like m : ( ( ) ( ) NORMSTR ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like total quasi_total complex-yielding V186() V187() ) Element of K6(K7(m : ( ( ) ( ) NORMSTR ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional FinSequence-membered ) Element of K6((REAL m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) = Z : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(m : ( ( ) ( ) NORMSTR ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in Z : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
it : ( ( Function-like quasi_total ) ( Relation-like m : ( ( ) ( ) NORMSTR ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like total quasi_total complex-yielding V186() V187() ) Element of K6(K7(m : ( ( ) ( ) NORMSTR ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) /. x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( Relation-like Function-like ) Element of Funcs ((REAL m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL n : ( ( ) ( ) Element of m : ( ( ) ( ) NORMSTR ) ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of REAL m : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) , REAL n : ( ( ) ( ) Element of m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) = diff (f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( Function-like quasi_total ) ( Relation-like REAL m : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL n : ( ( ) ( ) Element of m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like quasi_total ) Element of K6(K7((REAL m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL n : ( ( ) ( ) Element of m : ( ( ) ( ) NORMSTR ) ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) );
end;

theorem :: PDIFF_9:15
for m, n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for f, g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) is_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) & g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) is_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) holds
((f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) `| X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined Funcs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) , REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) -valued Function-like ) PartFunc of ,) /. x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( Relation-like Function-like ) Element of Funcs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) , REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) = (diff (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like total quasi_total V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) + (diff (g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like total quasi_total V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: PDIFF_9:16
for m, n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for f, g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) is_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) & g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) is_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) holds
((f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) `| X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined Funcs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) , REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) -valued Function-like ) PartFunc of ,) /. x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( Relation-like Function-like ) Element of Funcs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) , REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) = (diff (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like total quasi_total V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) - (diff (g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like total quasi_total V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: PDIFF_9:17
for m, n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,)
for r being ( ( ) ( V11() real ext-real ) Real) st f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) is_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) holds
( r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) holds
((r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) `| X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined Funcs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) , REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) -valued Function-like ) PartFunc of ,) /. x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( Relation-like Function-like ) Element of Funcs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) , REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) = r : ( ( ) ( V11() real ext-real ) Real) (#) (diff (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like total quasi_total V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: PDIFF_9:18
for j being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) ex p being ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) st
( p : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) = f : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) . <*1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) *> : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() V40(1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) & ( for r being ( ( ) ( V11() real ext-real ) Real)
for x being ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) st x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) = <*r : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) *> : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) FinSequence of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) holds
f : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) = r : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) * p : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) ) & ( for x being ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) holds ||.(f : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) .|| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) = ||.p : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) .|| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) * ||.x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) .|| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) ;

theorem :: PDIFF_9:19
for j being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) ex p being ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) st
( p : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) = f : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) . <*1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) *> : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() V40(1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) & ||.p : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) .|| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) = ||.f : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ;

theorem :: PDIFF_9:20
for j being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) )
for x being ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) holds ||.(f : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) .|| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) = ||.f : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) * ||.x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) .|| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ;

theorem :: PDIFF_9:21
for m, n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,)
for g 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,)
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for Y being ( ( ) ( ) Subset of ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) is open & g : ( ( 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) & X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) = Y : ( ( ) ( ) Subset of ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) )
for y being ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) st x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) & x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) = y : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) holds
partdiff (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) = (partdiff (g : ( ( 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) ,y : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) ,(REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) )) : ( ( non empty ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) . <*1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) *> : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() V40(1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ;

theorem :: PDIFF_9:22
for m, n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,)
for g 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,)
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for Y being ( ( ) ( ) Subset of ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) is open & g : ( ( 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) & X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) = Y : ( ( ) ( ) Subset of ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for x0, x1 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) )
for y0, y1 being ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) st x0 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) = y0 : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) & x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) = y1 : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) & x0 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) & x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) holds
|.(((f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) `partial| (X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like V33() V40(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) - ((f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) `partial| (X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x0 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like V33() V40(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) = ||.(((g : ( ( 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) `partial| (Y : ( ( ) ( ) Subset of ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) ,(REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) )) : ( ( non empty ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) ,(REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) )) : ( ( non empty ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. y1 : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) ,(REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) )) : ( ( non empty ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) - ((g : ( ( 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) `partial| (Y : ( ( ) ( ) Subset of ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) ,(REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) )) : ( ( non empty ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) ,(REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) )) : ( ( non empty ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. y0 : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) ,(REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) )) : ( ( non empty ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) ,(REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) )) : ( ( non empty ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ;

theorem :: PDIFF_9:23
for m, n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,)
for g 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,)
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for Y being ( ( ) ( ) Subset of ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) is open & g : ( ( 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) & X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) = Y : ( ( ) ( ) Subset of ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) `partial| (X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) iff ( g : ( ( 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) is_partial_differentiable_on Y : ( ( ) ( ) Subset of ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & g : ( ( 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) `partial| (Y : ( ( ) ( ) Subset of ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) ,(REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) )) : ( ( non empty ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) ,(REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) )) : ( ( non empty ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on Y : ( ( ) ( ) Subset of ) ) ) ;

theorem :: PDIFF_9:24
for m, n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,)
for g 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,)
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for Y being ( ( ) ( ) Subset of ) st X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) = Y : ( ( ) ( ) Subset of ) & X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) is open & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) = g : ( ( 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) holds
( ( for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) `partial| (X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ) ) iff ( g : ( ( 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) is_differentiable_on Y : ( ( ) ( ) Subset of ) & g : ( ( 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) `| Y : ( ( ) ( ) Subset of ) : ( ( 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) ,(REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) )) : ( ( non empty ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) ,(REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) )) : ( ( non empty ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on Y : ( ( ) ( ) Subset of ) ) ) ;

theorem :: PDIFF_9:25
for m, n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,)
for g 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,)
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for Y being ( ( ) ( ) Subset of ) st X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) is open & X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) c= dom f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Element of K6((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) & g : ( ( 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) & X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) = Y : ( ( ) ( ) Subset of ) holds
( g : ( ( 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) is_differentiable_on Y : ( ( ) ( ) Subset of ) & g : ( ( 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) `| Y : ( ( ) ( ) Subset of ) : ( ( 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) ,(REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) )) : ( ( non empty ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) ,(REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) )) : ( ( non empty ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on Y : ( ( ) ( ) Subset of ) iff ( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) is_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) & ( for x0 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) )
for r being ( ( ) ( V11() real ext-real ) Real) st x0 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) & 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional FinSequence-membered V160() V195() V196() V197() V198() V199() V200() V201() V202() V263() V264() V265() V266() V267() V268() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) holds
ex s being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional FinSequence-membered V160() V195() V196() V197() V198() V199() V200() V201() V202() V263() V264() V265() V266() V267() V268() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( ) ( V11() real ext-real ) Real) & ( for x1 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) & |.(x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) - x0 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) < s : ( ( ) ( V11() real ext-real ) Real) holds
for v being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) holds |.(((diff (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) ,x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like total quasi_total V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . v : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) - ((diff (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) ,x0 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like total quasi_total V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . v : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) <= r : ( ( ) ( V11() real ext-real ) Real) * |.v : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) ) ) ) ;

theorem :: PDIFF_9:26
for m, n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) st X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) is open & X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) c= dom f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Element of K6((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) holds
( ( for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) & i : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) <= m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) `partial| (X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ) ) iff ( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) is_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) & ( for x0 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) )
for r being ( ( ) ( V11() real ext-real ) Real) st x0 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) & 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional FinSequence-membered V160() V195() V196() V197() V198() V199() V200() V201() V202() V263() V264() V265() V266() V267() V268() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) holds
ex s being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional FinSequence-membered V160() V195() V196() V197() V198() V199() V200() V201() V202() V263() V264() V265() V266() V267() V268() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( ) ( V11() real ext-real ) Real) & ( for x1 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) & |.(x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) - x0 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) < s : ( ( ) ( V11() real ext-real ) Real) holds
for v being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) holds |.(((diff (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) ,x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like total quasi_total V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . v : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) - ((diff (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) ,x0 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like total quasi_total V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . v : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) <= r : ( ( ) ( V11() real ext-real ) Real) * |.v : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) ) ) ) ;

theorem :: PDIFF_9:27
for m, n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,)
for g 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) = g : ( ( 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) is_differentiable_on Z : ( ( ) ( ) set ) holds
f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) `| Z : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined Funcs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) , REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) -valued Function-like ) PartFunc of ,) = g : ( ( 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) `| Z : ( ( ) ( ) set ) : ( ( 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) ,(REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) )) : ( ( non empty ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) ,(REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) )) : ( ( non empty ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: PDIFF_9:28
for m, n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,)
for g 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,)
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for Y being ( ( ) ( ) Subset of ) st X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) = Y : ( ( ) ( ) Subset of ) & X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) is open & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) = g : ( ( 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) holds
( ( for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) `partial| (X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ) ) iff ( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) is_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) & g : ( ( 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) `| Y : ( ( ) ( ) Subset of ) : ( ( 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) ,(REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) )) : ( ( non empty ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) ,(REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) )) : ( ( non empty ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on Y : ( ( ) ( ) Subset of ) ) ) ;

theorem :: PDIFF_9:29
for m, n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f, g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,)
for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) is_continuous_in x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) & g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) is_continuous_in x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) ;

theorem :: PDIFF_9:30
for m, n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,)
for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) )
for r being ( ( ) ( V11() real ext-real ) Real) st f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) is_continuous_in x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) holds
r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ;

theorem :: PDIFF_9:31
for m, n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,)
for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) is_continuous_in x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) holds
- f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ;

theorem :: PDIFF_9:32
for m, n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,)
for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) is_continuous_in x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) holds
|.f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) .| : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ;

theorem :: PDIFF_9:33
for m, n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( ) ( ) set )
for f, g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) is_continuous_on Z : ( ( ) ( ) set ) & g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) is_continuous_on Z : ( ( ) ( ) set ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on Z : ( ( ) ( ) set ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on Z : ( ( ) ( ) set ) ) ;

theorem :: PDIFF_9:34
for m, n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( ) ( ) set )
for r being ( ( ) ( V11() real ext-real ) Real)
for f, g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) is_continuous_on Z : ( ( ) ( ) set ) holds
r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on Z : ( ( ) ( ) set ) ;

theorem :: PDIFF_9:35
for m, n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( ) ( ) set )
for f, g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) is_continuous_on Z : ( ( ) ( ) set ) holds
- f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on Z : ( ( ) ( ) set ) ;

theorem :: PDIFF_9:36
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,)
for x0 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) iff ( x0 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in dom f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Element of K6((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) & ( for r being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional FinSequence-membered V160() V195() V196() V197() V198() V199() V200() V201() V202() V263() V264() V265() V266() V267() V268() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) holds
ex s being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional FinSequence-membered V160() V195() V196() V197() V198() V199() V200() V201() V202() V263() V264() V265() V266() V267() V268() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( ) ( V11() real ext-real ) Real) & ( for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in dom f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Element of K6((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) & |.(x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) - x0 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) < s : ( ( ) ( V11() real ext-real ) Real) holds
|.((f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) /. x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) - (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) /. x0 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) .| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) < r : ( ( ) ( V11() real ext-real ) Real) ) ) ) ) ) ;

theorem :: PDIFF_9:37
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,)
for x0 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) iff <>* f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) ;

theorem :: PDIFF_9:38
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f, g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,)
for x0 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) & g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) ;

theorem :: PDIFF_9:39
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,)
for x0 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) )
for r being ( ( ) ( V11() real ext-real ) Real) st f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) holds
r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ;

theorem :: PDIFF_9:40
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,)
for x0 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) holds
|.f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) .| : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ;

theorem :: PDIFF_9:41
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f, g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,)
for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_continuous_in x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) & g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_continuous_in x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) holds
f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) (#) g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ;

definition
let m be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let Z be ( ( ) ( ) set ) ;
let f be ( ( Function-like ) ( Relation-like REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ;
pred f is_continuous_on Z means :: PDIFF_9:def 2
 for x0 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(m : ( ( ) ( ) NORMSTR ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st x0 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in Z : ( ( ) ( ) Element of m : ( ( ) ( ) NORMSTR ) ) holds
f : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( ) ( ) Element of m : ( ( ) ( ) NORMSTR ) ) : ( ( Function-like ) ( Relation-like REAL m : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ;
end;

theorem :: PDIFF_9:42
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,)
for g 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) = g : ( ( 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) holds
( ( Z : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Element of K6((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_continuous_on Z : ( ( ) ( ) set ) ) iff g : ( ( 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_continuous_on Z : ( ( ) ( ) set ) ) ;

theorem :: PDIFF_9:43
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,)
for g 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) = g : ( ( 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) & Z : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Element of K6((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_continuous_on Z : ( ( ) ( ) set ) iff for s being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty V2() ) set ) ) st rng s : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) c= Z : ( ( ) ( ) set ) & s : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty V2() ) set ) ) is convergent & lim s : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty V2() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) in Z : ( ( ) ( ) set ) holds
( g : ( ( 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) /* s : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty V2() ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like total quasi_total complex-yielding V186() V187() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is convergent & g : ( ( 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) /. (lim s : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty V2() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) = lim (g : ( ( 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) /* s : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty V2() ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like total quasi_total complex-yielding V186() V187() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) ;

theorem :: PDIFF_9:44
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,)
for g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) st <>* f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) holds
( ( Z : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Element of K6((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_continuous_on Z : ( ( ) ( ) set ) ) iff g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) is_continuous_on Z : ( ( ) ( ) set ) ) ;

theorem :: PDIFF_9:45
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) st Z : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Element of K6((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_continuous_on Z : ( ( ) ( ) set ) iff for x0 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) )
for r being ( ( ) ( V11() real ext-real ) Real) st x0 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in Z : ( ( ) ( ) set ) & 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional FinSequence-membered V160() V195() V196() V197() V198() V199() V200() V201() V202() V263() V264() V265() V266() V267() V268() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) holds
ex s being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional FinSequence-membered V160() V195() V196() V197() V198() V199() V200() V201() V202() V263() V264() V265() V266() V267() V268() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( ) ( V11() real ext-real ) Real) & ( for x1 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in Z : ( ( ) ( ) set ) & |.(x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) - x0 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) < s : ( ( ) ( V11() real ext-real ) Real) holds
|.((f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) /. x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) - (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) /. x0 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) .| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) < r : ( ( ) ( V11() real ext-real ) Real) ) ) ) ;

theorem :: PDIFF_9:46
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( ) ( ) set )
for f, g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_continuous_on Z : ( ( ) ( ) set ) & g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_continuous_on Z : ( ( ) ( ) set ) & Z : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Element of K6((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) & Z : ( ( ) ( ) set ) c= dom g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Element of K6((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on Z : ( ( ) ( ) set ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on Z : ( ( ) ( ) set ) ) ;

theorem :: PDIFF_9:47
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,)
for r being ( ( ) ( V11() real ext-real ) Real) st Z : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Element of K6((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_continuous_on Z : ( ( ) ( ) set ) holds
r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on Z : ( ( ) ( ) set ) ;

theorem :: PDIFF_9:48
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( ) ( ) set )
for f, g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_continuous_on Z : ( ( ) ( ) set ) & g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_continuous_on Z : ( ( ) ( ) set ) & Z : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Element of K6((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) & Z : ( ( ) ( ) set ) c= dom g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Element of K6((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) (#) g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on Z : ( ( ) ( ) set ) ;

theorem :: PDIFF_9:49
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,)
for g 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) = g : ( ( 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) holds
( ( Z : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Element of K6((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_continuous_on Z : ( ( ) ( ) set ) ) iff g : ( ( 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V182() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_continuous_on Z : ( ( ) ( ) set ) ) ;

theorem :: PDIFF_9:50
for m, n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( ) ( ) set )
for f, g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) is_continuous_on Z : ( ( ) ( ) set ) holds
|.f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) .| : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on Z : ( ( ) ( ) set ) ;

theorem :: PDIFF_9:51
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f, g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,)
for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_differentiable_in x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) & g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_differentiable_in x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_in x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) & diff ((f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like total quasi_total complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) = (diff (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like total quasi_total complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) + (diff (g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like total quasi_total complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like total quasi_total complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_in x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) & diff ((f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like total quasi_total complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) = (diff (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like total quasi_total complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) - (diff (g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like total quasi_total complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like total quasi_total complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: PDIFF_9:52
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,)
for r being ( ( ) ( V11() real ext-real ) Real)
for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_differentiable_in x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) holds
( r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_in x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) & diff ((r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like total quasi_total complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) = r : ( ( ) ( V11() real ext-real ) Real) (#) (diff (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like total quasi_total complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like total quasi_total complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) ) ;

definition
let Z be ( ( ) ( ) set ) ;
let m be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ;
pred f is_differentiable_on Z means :: PDIFF_9:def 3
 for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(m : ( ( ) ( ) Element of Z : ( ( ) ( ) NORMSTR ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( ) ( ) Element of Z : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in Z : ( ( ) ( ) NORMSTR ) holds
f : ( ( Function-like quasi_total ) ( Relation-like K7(Z : ( ( ) ( ) NORMSTR ) ,Z : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined Z : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(Z : ( ( ) ( ) NORMSTR ) ,Z : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,Z : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( ) ( ) NORMSTR ) : ( ( Function-like ) ( Relation-like REAL m : ( ( ) ( ) Element of Z : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL m : ( ( ) ( ) Element of Z : ( ( ) ( ) NORMSTR ) ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_in x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ;
end;

theorem :: PDIFF_9:53
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,)
for g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) st <>* f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) holds
( ( Z : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Element of K6((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_differentiable_on Z : ( ( ) ( ) set ) ) iff g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) is_differentiable_on Z : ( ( ) ( ) set ) ) ;

theorem :: PDIFF_9:54
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) st X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) c= dom f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Element of K6((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) & X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) is open holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) iff for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) holds
f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_differentiable_in x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) ;

theorem :: PDIFF_9:55
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) st X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) c= dom f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Element of K6((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) holds
X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) is open ;

definition
let m be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let Z be ( ( ) ( ) set ) ;
let f be ( ( Function-like ) ( Relation-like REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ;
assume Z : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Element of K6((REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) ;
func f `| Z -> ( ( Function-like ) ( Relation-like REAL m : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined Funcs ((REAL m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) FUNCTION_DOMAIN of REAL m : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) , REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) means :: PDIFF_9:def 4
( dom it : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like functional FinSequence-membered ) Element of K6((REAL m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) = Z : ( ( ) ( ) Element of m : ( ( ) ( ) NORMSTR ) ) & ( for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(m : ( ( ) ( ) NORMSTR ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in Z : ( ( ) ( ) Element of m : ( ( ) ( ) NORMSTR ) ) holds
it : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( Relation-like Function-like complex-yielding V186() V187() ) Element of Funcs ((REAL m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) FUNCTION_DOMAIN of REAL m : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) , REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) = diff (f : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( Function-like quasi_total ) ( Relation-like REAL m : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like total quasi_total complex-yielding V186() V187() ) Element of K6(K7((REAL m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) ) );
end;

theorem :: PDIFF_9:56
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,)
for g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) st <>* f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) & X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) c= dom f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Element of K6((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) holds
( g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) is_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) holds
(f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) `| X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined Funcs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) FUNCTION_DOMAIN of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) , REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) /. x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( Relation-like Function-like complex-yielding V186() V187() ) Element of Funcs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) FUNCTION_DOMAIN of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) , REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) = (proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) 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 V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like total quasi_total complex-yielding V186() V187() ) Element of K6(K7((REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) * ((g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) `| X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined Funcs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) , REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) -valued Function-like ) PartFunc of ,) /. x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( ) ( Relation-like Function-like ) Element of Funcs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) , REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( Relation-like ) ( Relation-like Function-like complex-yielding V186() V187() ) set ) ) ) ;

theorem :: PDIFF_9:57
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for f, g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) st X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) c= dom f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Element of K6((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) & X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) c= dom g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Element of K6((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) & g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) holds
((f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) `| X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined Funcs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) FUNCTION_DOMAIN of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) , REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) /. x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( Relation-like Function-like complex-yielding V186() V187() ) Element of Funcs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) FUNCTION_DOMAIN of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) , REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) = ((f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) `| X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined Funcs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) FUNCTION_DOMAIN of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) , REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) /. x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( ) ( Relation-like Function-like complex-yielding V186() V187() ) Element of Funcs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) FUNCTION_DOMAIN of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) , REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) + ((g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) `| X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined Funcs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) FUNCTION_DOMAIN of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) , REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) /. x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( ) ( Relation-like Function-like complex-yielding V186() V187() ) Element of Funcs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) FUNCTION_DOMAIN of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) , REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-yielding V186() V187() ) set ) ) ) ;

theorem :: PDIFF_9:58
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for f, g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) st X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) c= dom f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Element of K6((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) & X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) c= dom g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Element of K6((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) & g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) holds
((f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) `| X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined Funcs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) FUNCTION_DOMAIN of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) , REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) /. x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( Relation-like Function-like complex-yielding V186() V187() ) Element of Funcs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) FUNCTION_DOMAIN of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) , REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) = ((f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) `| X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined Funcs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) FUNCTION_DOMAIN of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) , REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) /. x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( ) ( Relation-like Function-like complex-yielding V186() V187() ) Element of Funcs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) FUNCTION_DOMAIN of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) , REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) - ((g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) `| X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined Funcs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) FUNCTION_DOMAIN of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) , REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) /. x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( ) ( Relation-like Function-like complex-yielding V186() V187() ) Element of Funcs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) FUNCTION_DOMAIN of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) , REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-yielding V186() V187() ) set ) ) ) ;

theorem :: PDIFF_9:59
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,)
for r being ( ( ) ( V11() real ext-real ) Real) st X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) c= dom f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Element of K6((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) holds
( r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) holds
((r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) `| X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined Funcs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) FUNCTION_DOMAIN of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) , REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) /. x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( Relation-like Function-like complex-yielding V186() V187() ) Element of Funcs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) FUNCTION_DOMAIN of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) , REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) = r : ( ( ) ( V11() real ext-real ) Real) (#) ((f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) `| X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined Funcs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) FUNCTION_DOMAIN of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) , REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) /. x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( ) ( Relation-like Function-like complex-yielding V186() V187() ) Element of Funcs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) FUNCTION_DOMAIN of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) , REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-yielding V186() V187() ) set ) ) ) ;

definition
let m be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let Z be ( ( ) ( ) set ) ;
let i be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ;
pred f is_partial_differentiable_on Z,i means :: PDIFF_9:def 5
( Z : ( ( ) ( ) Element of m : ( ( ) ( ) NORMSTR ) ) c= dom f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like functional FinSequence-membered ) Element of K6((REAL m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(m : ( ( ) ( ) NORMSTR ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in Z : ( ( ) ( ) Element of m : ( ( ) ( ) NORMSTR ) ) holds
f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( ) ( ) Element of m : ( ( ) ( ) NORMSTR ) ) : ( ( Function-like ) ( Relation-like REAL m : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_partial_differentiable_in x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ,i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) );
end;

definition
let m be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let Z be ( ( ) ( ) set ) ;
let i be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ;
assume f : ( ( Function-like ) ( Relation-like REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_on Z : ( ( ) ( ) set ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ;
func f `partial| (Z,i) -> ( ( Function-like ) ( Relation-like REAL m : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) means :: PDIFF_9:def 6
( dom it : ( ( Function-like quasi_total ) ( Relation-like m : ( ( ) ( ) NORMSTR ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like total quasi_total complex-yielding V186() V187() ) Element of K6(K7(m : ( ( ) ( ) NORMSTR ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional FinSequence-membered ) Element of K6((REAL m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) = Z : ( ( ) ( ) Element of m : ( ( ) ( ) NORMSTR ) ) & ( for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(m : ( ( ) ( ) NORMSTR ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in Z : ( ( ) ( ) Element of m : ( ( ) ( ) NORMSTR ) ) holds
it : ( ( Function-like quasi_total ) ( Relation-like m : ( ( ) ( ) NORMSTR ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like total quasi_total complex-yielding V186() V187() ) Element of K6(K7(m : ( ( ) ( ) NORMSTR ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) /. x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) = partdiff (f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ,i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) );
end;

theorem :: PDIFF_9:60
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) st X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) is open & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) iff ( X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) c= dom f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Element of K6((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) holds
f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_in x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ) ;

theorem :: PDIFF_9:61
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,)
for g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) st <>* f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) & X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) is open & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) iff g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: PDIFF_9:62
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,)
for g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) st <>* f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) & X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) is open & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) `partial| (X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_continuous_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) iff g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) PartFunc of ,) `partial| (X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -valued Function-like V269() V270() V271() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,(REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ) ;

theorem :: PDIFF_9:63
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) st X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) is open & X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) c= dom f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Element of K6((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( ) set ) ) holds
( ( for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) & i : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) <= m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) `partial| (X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_continuous_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ) ) iff ( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) & ( for x0 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) )
for r being ( ( ) ( V11() real ext-real ) Real) st x0 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) & 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional FinSequence-membered V160() V195() V196() V197() V198() V199() V200() V201() V202() V263() V264() V265() V266() V267() V268() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) holds
ex s being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional FinSequence-membered V160() V195() V196() V197() V198() V199() V200() V201() V202() V263() V264() V265() V266() V267() V268() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( ) ( V11() real ext-real ) Real) & ( for x1 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) & |.(x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) - x0 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) < s : ( ( ) ( V11() real ext-real ) Real) holds
for v being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) holds |.(((diff (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ,x1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like total quasi_total complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) . v : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) - ((diff (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ,x0 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like total quasi_total complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) . v : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) .| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) <= r : ( ( ) ( V11() real ext-real ) Real) * |.v : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) ) ) ) ;

theorem :: PDIFF_9:64
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f, g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,)
for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_in x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_in x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) (#) g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_partial_differentiable_in x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & partdiff ((f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) (#) g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) = ((partdiff (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) * (g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) . x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) + ((f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) . x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) * (partdiff (g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ;

theorem :: PDIFF_9:65
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for f, g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) st X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) is open & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) `partial| (X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) = (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) `partial| (X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) + (g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) `partial| (X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) holds
((f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) `partial| (X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) /. x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) = (partdiff (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) + (partdiff (g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) ;

theorem :: PDIFF_9:66
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for f, g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) st X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) is open & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) `partial| (X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) = (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) `partial| (X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) - (g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) `partial| (X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) holds
((f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) `partial| (X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) /. x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) = (partdiff (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) - (partdiff (g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) ;

theorem :: PDIFF_9:67
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for r being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) st X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) is open & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & (r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) `partial| (X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) = r : ( ( ) ( V11() real ext-real ) Real) (#) (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) `partial| (X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) holds
((r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) `partial| (X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) /. x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) = r : ( ( ) ( V11() real ext-real ) Real) * (partdiff (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) ;

theorem :: PDIFF_9:68
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for f, g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) st X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) is open & 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) (#) g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) (#) g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) `partial| (X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) = ((f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) `partial| (X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) (#) g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) + (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) (#) (g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) `partial| (X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) st x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) in X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) holds
((f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) (#) g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) `partial| (X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) /. x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) = ((partdiff (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) * (g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) . x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) + ((f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) . x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) * (partdiff (g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) ) ;

begin

definition
let m be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let Z be ( ( ) ( ) set ) ;
let I be ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ;
func PartDiffSeq (f,Z,I) -> ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs ((REAL m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) set ) -valued Function-like total quasi_total V269() V270() V271() ) Functional_Sequence of ( ( ) ( non empty functional V263() V264() V265() ) set ) ,(REAL m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) means :: PDIFF_9:def 7
( it : ( ( Function-like quasi_total ) ( Relation-like m : ( ( ) ( ) NORMSTR ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like total quasi_total complex-yielding V186() V187() ) Element of K6(K7(m : ( ( ) ( ) NORMSTR ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional FinSequence-membered V160() V195() V196() V197() V198() V199() V200() V201() V202() V263() V264() V265() V266() V267() V268() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V11() real ext-real Relation-like REAL m : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) = f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ( for i being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) number ) holds it : ( ( Function-like quasi_total ) ( Relation-like m : ( ( ) ( ) NORMSTR ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like total quasi_total complex-yielding V186() V187() ) Element of K6(K7(m : ( ( ) ( ) NORMSTR ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) . (i : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) number ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V11() real ext-real Relation-like REAL m : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) = (it : ( ( Function-like quasi_total ) ( Relation-like m : ( ( ) ( ) NORMSTR ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like total quasi_total complex-yielding V186() V187() ) Element of K6(K7(m : ( ( ) ( ) NORMSTR ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) . i : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) number ) ) : ( ( Function-like ) ( V11() real ext-real Relation-like REAL m : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) `partial| (Z : ( ( ) ( ) Element of m : ( ( ) ( ) NORMSTR ) ) ,(I : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. (i : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) number ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL m : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) );
end;

definition
let m be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let Z be ( ( ) ( ) set ) ;
let I be ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ;
pred f is_partial_differentiable_on Z,I means :: PDIFF_9:def 8
 for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) st i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= (len I : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real V160() V195() ) Element of INT : ( ( ) ( non empty V33() V196() V197() V198() V199() V200() V202() ) set ) ) holds
(PartDiffSeq (f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( ) ( ) Element of m : ( ( ) ( ) NORMSTR ) ) ,I : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs ((REAL m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) set ) -valued Function-like total quasi_total V269() V270() V271() ) Functional_Sequence of ( ( ) ( non empty functional V263() V264() V265() ) set ) ,(REAL m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL m : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_partial_differentiable_on Z : ( ( ) ( ) Element of m : ( ( ) ( ) NORMSTR ) ) ,I : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ;
end;

definition
let m be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let Z be ( ( ) ( ) set ) ;
let I be ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ;
func f `partial| (Z,I) -> ( ( Function-like ) ( Relation-like REAL m : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) equals :: PDIFF_9:def 9
(PartDiffSeq (f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( ) ( ) Element of m : ( ( ) ( ) NORMSTR ) ) ,I : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs ((REAL m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) set ) -valued Function-like total quasi_total V269() V270() V271() ) Functional_Sequence of ( ( ) ( non empty functional V263() V264() V265() ) set ) ,(REAL m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) . (len I : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL m : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) ;
end;

theorem :: PDIFF_9:69
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for I being ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f, g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) st X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) is open & rng I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V196() V197() V198() V199() V200() V201() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) c= Seg m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) V196() V197() V198() V199() V200() V201() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) st i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= (len I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real V160() V195() ) Element of INT : ( ( ) ( non empty V33() V196() V197() V198() V199() V200() V202() ) set ) ) holds
( (PartDiffSeq ((f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) ,X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) set ) -valued Function-like total quasi_total V269() V270() V271() ) Functional_Sequence of ( ( ) ( non empty functional V263() V264() V265() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) /. (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & (PartDiffSeq ((f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) ,X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) set ) -valued Function-like total quasi_total V269() V270() V271() ) Functional_Sequence of ( ( ) ( non empty functional V263() V264() V265() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) = ((PartDiffSeq (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ,X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) set ) -valued Function-like total quasi_total V269() V270() V271() ) Functional_Sequence of ( ( ) ( non empty functional V263() V264() V265() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) + ((PartDiffSeq (g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ,X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) set ) -valued Function-like total quasi_total V269() V270() V271() ) Functional_Sequence of ( ( ) ( non empty functional V263() V264() V265() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: PDIFF_9:70
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for I being ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f, g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) st X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) is open & rng I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V196() V197() V198() V199() V200() V201() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) c= Seg m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) V196() V197() V198() V199() V200() V201() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) `partial| (X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) = (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) `partial| (X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) + (g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) `partial| (X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: PDIFF_9:71
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for I being ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f, g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) st X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) is open & rng I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V196() V197() V198() V199() V200() V201() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) c= Seg m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) V196() V197() V198() V199() V200() V201() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) st i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= (len I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real V160() V195() ) Element of INT : ( ( ) ( non empty V33() V196() V197() V198() V199() V200() V202() ) set ) ) holds
( (PartDiffSeq ((f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) ,X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) set ) -valued Function-like total quasi_total V269() V270() V271() ) Functional_Sequence of ( ( ) ( non empty functional V263() V264() V265() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) /. (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & (PartDiffSeq ((f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) ,X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) set ) -valued Function-like total quasi_total V269() V270() V271() ) Functional_Sequence of ( ( ) ( non empty functional V263() V264() V265() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) = ((PartDiffSeq (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ,X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) set ) -valued Function-like total quasi_total V269() V270() V271() ) Functional_Sequence of ( ( ) ( non empty functional V263() V264() V265() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) - ((PartDiffSeq (g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ,X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) set ) -valued Function-like total quasi_total V269() V270() V271() ) Functional_Sequence of ( ( ) ( non empty functional V263() V264() V265() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: PDIFF_9:72
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for I being ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f, g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) st X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) is open & rng I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V196() V197() V198() V199() V200() V201() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) c= Seg m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) V196() V197() V198() V199() V200() V201() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) `partial| (X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) = (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) `partial| (X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) - (g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) `partial| (X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: PDIFF_9:73
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for r being ( ( ) ( V11() real ext-real ) Real)
for I being ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) st X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) is open & rng I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V196() V197() V198() V199() V200() V201() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) c= Seg m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) V196() V197() V198() V199() V200() V201() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) st i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= (len I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real V160() V195() ) Element of INT : ( ( ) ( non empty V33() V196() V197() V198() V199() V200() V202() ) set ) ) holds
( (PartDiffSeq ((r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) ,X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) set ) -valued Function-like total quasi_total V269() V270() V271() ) Functional_Sequence of ( ( ) ( non empty functional V263() V264() V265() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) /. (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & (PartDiffSeq ((r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) ,X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) set ) -valued Function-like total quasi_total V269() V270() V271() ) Functional_Sequence of ( ( ) ( non empty functional V263() V264() V265() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) = r : ( ( ) ( V11() real ext-real ) Real) (#) ((PartDiffSeq (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ,X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs ((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( non empty functional V263() V264() V265() ) set ) -valued Function-like total quasi_total V269() V270() V271() ) Functional_Sequence of ( ( ) ( non empty functional V263() V264() V265() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: PDIFF_9:74
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for r being ( ( ) ( V11() real ext-real ) Real)
for I being ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) st X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) is open & rng I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V196() V197() V198() V199() V200() V201() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) c= Seg m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) V196() V197() V198() V199() V200() V201() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & (r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) `partial| (X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) = r : ( ( ) ( V11() real ext-real ) Real) (#) (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) `partial| (X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) ) ;

definition
let m be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ;
let k be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let Z be ( ( ) ( ) set ) ;
pred f is_partial_differentiable_up_to_order k,Z means :: PDIFF_9:def 10
 for I being ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) st len I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= k : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & rng I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V196() V197() V198() V199() V200() V201() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) c= Seg m : ( ( ) ( ) NORMSTR ) : ( ( ) ( V196() V197() V198() V199() V200() V201() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) holds
f : ( ( ) ( ) Element of m : ( ( ) ( ) NORMSTR ) ) is_partial_differentiable_on Z : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ;
end;

theorem :: PDIFF_9:75
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,)
for I, G being ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_on Z : ( ( ) ( ) set ) ,G : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ^ I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) iff ( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_on Z : ( ( ) ( ) set ) ,G : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) `partial| (Z : ( ( ) ( ) set ) ,G : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_on Z : ( ( ) ( ) set ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ;

theorem :: PDIFF_9:76
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_on Z : ( ( ) ( ) set ) ,<*i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) *> : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() V40(1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) iff f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_on Z : ( ( ) ( ) set ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: PDIFF_9:77
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) holds f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) `partial| (Z : ( ( ) ( ) set ) ,<*i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) *> : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() V40(1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) `partial| (Z : ( ( ) ( ) set ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) ;

theorem :: PDIFF_9:78
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for i, j being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,)
for I being ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) st f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_up_to_order i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) + j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( ) ( ) set ) & rng I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V196() V197() V198() V199() V200() V201() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) c= Seg m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V33() V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) V196() V197() V198() V199() V200() V201() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) & len I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) = j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) holds
f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) `partial| (Z : ( ( ) ( ) set ) ,I : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V186() V187() V188() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_up_to_order i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( ) ( ) set ) ;

theorem :: PDIFF_9:79
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for i, j being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_up_to_order i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( ) ( ) set ) & j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) holds
f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_up_to_order j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( ) ( ) set ) ;

theorem :: PDIFF_9:80
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for f, g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) st X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) is open & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_up_to_order i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ,X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) & g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_up_to_order i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ,X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_partial_differentiable_up_to_order i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ,X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_partial_differentiable_up_to_order i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ,X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ) ;

theorem :: PDIFF_9:81
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,)
for r being ( ( ) ( V11() real ext-real ) Real) st X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) is open & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_up_to_order i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ,X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) holds
r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_partial_differentiable_up_to_order i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ,X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ;

theorem :: PDIFF_9:82
for m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for X being ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) st X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) is open holds
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) )
for f, g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_up_to_order i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ,X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) & g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) is_partial_differentiable_up_to_order i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ,X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) holds
f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) (#) g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) -valued Function-like complex-yielding V186() V187() ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered V263() V264() V265() ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) ,REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( complex-yielding V186() V187() ) set ) ) : ( ( ) ( ) set ) ) is_partial_differentiable_up_to_order i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V160() V195() V196() V197() V198() V199() V200() V201() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V196() V197() V198() V199() V200() V201() V202() ) Element of K6(REAL : ( ( ) ( non empty V33() V196() V197() V198() V202() ) set ) ) : ( ( ) ( ) set ) ) ) ,X : ( ( ) ( functional FinSequence-membered V263() V264() V265() ) Subset of ( ( ) ( ) set ) ) ;