:: NDIFF_1 semantic presentation

begin

theorem :: NDIFF_1:1
for S being ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for x0 being ( ( ) ( ) Point of ( ( ) ( V1() ) set ) )
for N1, N2 being ( ( ) ( ) Neighbourhood of x0 : ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) ) ex N being ( ( ) ( ) Neighbourhood of x0 : ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) ) st
( N : ( ( ) ( ) Neighbourhood of b2 : ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) ) c= N1 : ( ( ) ( ) Neighbourhood of b2 : ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) ) & N : ( ( ) ( ) Neighbourhood of b2 : ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) ) c= N2 : ( ( ) ( ) Neighbourhood of b2 : ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) ) ) ;

theorem :: NDIFF_1:2
for S being ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) Subset of ) st X : ( ( ) ( ) Subset of ) is open holds
for r being ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) st r : ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) in X : ( ( ) ( ) Subset of ) holds
ex N being ( ( ) ( ) Neighbourhood of r : ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) ) st N : ( ( ) ( ) Neighbourhood of b3 : ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) ) c= X : ( ( ) ( ) Subset of ) ;

theorem :: NDIFF_1:3
for S being ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) Subset of ) st X : ( ( ) ( ) Subset of ) is open holds
for r being ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) st r : ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) in X : ( ( ) ( ) Subset of ) holds
ex g being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) < g : ( ( ) ( V11() real ext-real ) Real) & { y : ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) where y is ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) : ||.(y : ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) - r : ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) < g : ( ( ) ( V11() real ext-real ) Real) } c= X : ( ( ) ( ) Subset of ) ) ;

theorem :: NDIFF_1:4
for S being ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) Subset of ) st ( for r being ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) st r : ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) in X : ( ( ) ( ) Subset of ) holds
ex N being ( ( ) ( ) Neighbourhood of r : ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) ) st N : ( ( ) ( ) Neighbourhood of b3 : ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) ) c= X : ( ( ) ( ) Subset of ) ) holds
X : ( ( ) ( ) Subset of ) is open ;

theorem :: NDIFF_1:5
for S being ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) Subset of ) holds
( ( for r being ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) st r : ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) in X : ( ( ) ( ) Subset of ) holds
ex N being ( ( ) ( ) Neighbourhood of r : ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) ) st N : ( ( ) ( ) Neighbourhood of b3 : ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) ) c= X : ( ( ) ( ) Subset of ) ) iff X : ( ( ) ( ) Subset of ) is open ) ;

definition
let X be ( ( ) ( ) set ) ;
let S be ( ( ) ( ) ZeroStr ) ;
let f be ( ( Function-like quasi_total ) ( V16() V19(X : ( ( ) ( ) set ) ) V20( the carrier of S : ( ( ) ( ) ZeroStr ) : ( ( ) ( ) set ) ) Function-like quasi_total ) Function of X : ( ( ) ( ) set ) , ( ( ) ( ) set ) ) ;
redefine attr f is non-zero means :: NDIFF_1:def 1
 rng f : ( ( Function-like quasi_total ) ( V16() V19(K7(X : ( ( ) ( ) Normed_AlgebraStr ) ,X : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ) V20(X : ( ( ) ( ) Normed_AlgebraStr ) ) Function-like quasi_total ) Element of K6(K7(K7(X : ( ( ) ( ) Normed_AlgebraStr ) ,X : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the carrier of S : ( ( Function-like quasi_total ) ( V16() V19(K7(X : ( ( ) ( ) Normed_AlgebraStr ) ,X : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ) V20(X : ( ( ) ( ) Normed_AlgebraStr ) ) Function-like quasi_total ) Element of K6(K7(K7(X : ( ( ) ( ) Normed_AlgebraStr ) ,X : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) c= NonZero S : ( ( Function-like quasi_total ) ( V16() V19(K7(X : ( ( ) ( ) Normed_AlgebraStr ) ,X : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ) V20(X : ( ( ) ( ) Normed_AlgebraStr ) ) Function-like quasi_total ) Element of K6(K7(K7(X : ( ( ) ( ) Normed_AlgebraStr ) ,X : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the carrier of S : ( ( Function-like quasi_total ) ( V16() V19(K7(X : ( ( ) ( ) Normed_AlgebraStr ) ,X : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ) V20(X : ( ( ) ( ) Normed_AlgebraStr ) ) Function-like quasi_total ) Element of K6(K7(K7(X : ( ( ) ( ) Normed_AlgebraStr ) ,X : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;
end;

theorem :: NDIFF_1:6
for S being ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) holds
( seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) is non-zero iff for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) holds
seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( ) set ) <> 0. S : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) ) ;

theorem :: NDIFF_1:7
for S being ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) holds
( seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) is non-zero iff for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) holds seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) <> 0. S : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) ) ;

definition
let RNS be ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) ;
let S be ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of RNS : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) ;
let a be ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ;
func a (#) S -> ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of RNS : ( ( ) ( ) Normed_AlgebraStr ) : ( ( ) ( ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( ) set ) ) means :: NDIFF_1:def 2
 for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) holds it : ( ( Function-like quasi_total ) ( V16() V19(K7(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ,RNS : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ) V20(RNS : ( ( ) ( ) Normed_AlgebraStr ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ,RNS : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of RNS : ( ( ) ( ) Normed_AlgebraStr ) : ( ( ) ( ) set ) ) = (a : ( ( Function-like quasi_total ) ( V16() V19(K7(RNS : ( ( ) ( ) Normed_AlgebraStr ) ,RNS : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ) V20(RNS : ( ( ) ( ) Normed_AlgebraStr ) ) Function-like quasi_total ) Element of K6(K7(K7(RNS : ( ( ) ( ) Normed_AlgebraStr ) ,RNS : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) * (S : ( ( Function-like quasi_total ) ( V16() V19(K7(RNS : ( ( ) ( ) Normed_AlgebraStr ) ,RNS : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ) V20(RNS : ( ( ) ( ) Normed_AlgebraStr ) ) Function-like quasi_total ) Element of K6(K7(K7(RNS : ( ( ) ( ) Normed_AlgebraStr ) ,RNS : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of RNS : ( ( ) ( ) Normed_AlgebraStr ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of RNS : ( ( ) ( ) Normed_AlgebraStr ) : ( ( ) ( ) set ) ) ;
end;

definition
let RNS be ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RealLinearSpace) ;
let z be ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) ;
let a be ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ;
func a * z -> ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of RNS : ( ( ) ( ) Normed_AlgebraStr ) : ( ( ) ( ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( ) set ) ) means :: NDIFF_1:def 3
 for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) holds it : ( ( Function-like quasi_total ) ( V16() V19(K7(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ,RNS : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ) V20(RNS : ( ( ) ( ) Normed_AlgebraStr ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ,RNS : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of RNS : ( ( ) ( ) Normed_AlgebraStr ) : ( ( ) ( ) set ) ) = (a : ( ( Function-like quasi_total ) ( V16() V19(K7(RNS : ( ( ) ( ) Normed_AlgebraStr ) ,RNS : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ) V20(RNS : ( ( ) ( ) Normed_AlgebraStr ) ) Function-like quasi_total ) Element of K6(K7(K7(RNS : ( ( ) ( ) Normed_AlgebraStr ) ,RNS : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) * z : ( ( Function-like quasi_total ) ( V16() V19(K7(RNS : ( ( ) ( ) Normed_AlgebraStr ) ,RNS : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ) V20(RNS : ( ( ) ( ) Normed_AlgebraStr ) ) Function-like quasi_total ) Element of K6(K7(K7(RNS : ( ( ) ( ) Normed_AlgebraStr ) ,RNS : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ,RNS : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of RNS : ( ( ) ( ) Normed_AlgebraStr ) : ( ( ) ( ) set ) ) ;
end;

theorem :: NDIFF_1:8
for S being ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) )
for rseq1, rseq2 being ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) holds (rseq1 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) + rseq2 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) = (rseq1 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) + (rseq2 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: NDIFF_1:9
for S being ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for rseq being ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence)
for seq1, seq2 being ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) holds rseq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) (#) (seq1 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) + seq2 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) = (rseq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq1 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) + (rseq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq2 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: NDIFF_1:10
for r being ( ( ) ( V11() real ext-real ) Real)
for S being ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) )
for rseq being ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) holds r : ( ( ) ( V11() real ext-real ) Real) * (rseq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = rseq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) (#) (r : ( ( ) ( V11() real ext-real ) Real) * seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) ;

theorem :: NDIFF_1:11
for S being ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) )
for rseq1, rseq2 being ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) holds (rseq1 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) - rseq2 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) = (rseq1 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) - (rseq2 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: NDIFF_1:12
for S being ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for rseq being ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence)
for seq1, seq2 being ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) holds rseq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) (#) (seq1 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) - seq2 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) = (rseq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq1 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) - (rseq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq2 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: NDIFF_1:13
for S being ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for rseq being ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence)
for seq being ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) st rseq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) is convergent holds
rseq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) is convergent ;

theorem :: NDIFF_1:14
for S being ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for rseq being ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence)
for seq being ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) st rseq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) is convergent holds
lim (rseq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) = (lim rseq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) * (lim seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) ;

theorem :: NDIFF_1:15
for k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) )
for S being ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for seq, seq1 being ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) holds (seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) + seq1 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ^\ k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) subsequence of b3 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) + b4 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) = (seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) ^\ k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) subsequence of b3 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) ) + (seq1 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) ^\ k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) subsequence of b4 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: NDIFF_1:16
for k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) )
for S being ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for seq, seq1 being ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) holds (seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) - seq1 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ^\ k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) subsequence of b3 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) - b4 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) = (seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) ^\ k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) subsequence of b3 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) ) - (seq1 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) ^\ k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) subsequence of b4 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: NDIFF_1:17
for k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) )
for S being ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) st seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) is non-zero holds
seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) ^\ k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) subsequence of b3 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) ) is non-zero ;

definition
let S be ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ;
let x be ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) ;
let IT be ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of S : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) ;
attr IT is x -convergent means :: NDIFF_1:def 4
( IT : ( ( Function-like quasi_total ) ( V16() V19(K7(S : ( ( ) ( ) Normed_AlgebraStr ) ,S : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ) V20(S : ( ( ) ( ) Normed_AlgebraStr ) ) Function-like quasi_total ) Element of K6(K7(K7(S : ( ( ) ( ) Normed_AlgebraStr ) ,S : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ,S : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is convergent & lim IT : ( ( Function-like quasi_total ) ( V16() V19(K7(S : ( ( ) ( ) Normed_AlgebraStr ) ,S : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ) V20(S : ( ( ) ( ) Normed_AlgebraStr ) ) Function-like quasi_total ) Element of K6(K7(K7(S : ( ( ) ( ) Normed_AlgebraStr ) ,S : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ,S : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of S : ( ( ) ( ) Normed_AlgebraStr ) : ( ( ) ( ) set ) ) = x : ( ( Function-like quasi_total ) ( V16() V19(K7(S : ( ( ) ( ) Normed_AlgebraStr ) ,S : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ) V20(S : ( ( ) ( ) Normed_AlgebraStr ) ) Function-like quasi_total ) Element of K6(K7(K7(S : ( ( ) ( ) Normed_AlgebraStr ) ,S : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ,S : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) );
end;

theorem :: NDIFF_1:18
for X being ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) st seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) is constant holds
( seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) is convergent & ( for k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) holds lim seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) = seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) . k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) ) ) ;

theorem :: NDIFF_1:19
for S being ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) )
for x0 being ( ( ) ( ) Point of ( ( ) ( V1() ) set ) )
for r being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) holds seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) = (1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) / (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) * x0 : ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) ) holds
seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) is convergent ;

theorem :: NDIFF_1:20
for S being ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) )
for x0 being ( ( ) ( ) Point of ( ( ) ( V1() ) set ) )
for r being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) holds seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) = (1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) / (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) * x0 : ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) ) holds
lim seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) = 0. S : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) ;

registration
let S be ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ;
cluster V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total non-zero 0. S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V68(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) ) Element of the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) -convergent for ( ( ) ( ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;
end;

theorem :: NDIFF_1:21
for S being ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for a being ( ( non-zero Function-like quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() V16() non-zero V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence)
for z being ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) st z : ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) <> 0. S : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) holds
( a : ( ( non-zero Function-like quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() V16() non-zero V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) * z : ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) is non-zero & a : ( ( non-zero Function-like quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() V16() non-zero V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) * z : ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) is 0. S : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of b1 : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) -convergent ) ;

theorem :: NDIFF_1:22
for S being ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for Y being ( ( ) ( ) Subset of ) holds
( ( for r being ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) holds
( r : ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) in Y : ( ( ) ( ) Subset of ) iff r : ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) in the carrier of S : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) ) iff Y : ( ( ) ( ) Subset of ) = the carrier of S : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) ;

registration
let S be ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ;
cluster V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) Function-like constant total quasi_total for ( ( ) ( ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;
end;

definition
let S, T be ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ;
let IT be ( ( Function-like ) ( V16() V19( the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ;
attr IT is RestFunc-like means :: NDIFF_1:def 5
( IT : ( ( Function-like quasi_total ) ( V16() V19(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) V20(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is total & ( for h being ( ( Function-like quasi_total non-zero 0. S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V68(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) ) Element of the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) -convergent ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total non-zero 0. S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V68(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) ) Element of the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) -convergent ) sequence of ( ( ) ( V1() non trivial ) set ) ) holds
( (||.h : ( ( Function-like quasi_total non-zero 0. S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) -convergent ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total non-zero 0. S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) -convergent ) sequence of ( ( ) ( V1() non trivial ) set ) ) .|| : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ") : ( ( Function-like ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) (IT : ( ( Function-like quasi_total ) ( V16() V19(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) V20(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /* h : ( ( Function-like quasi_total non-zero 0. S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) -convergent ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total non-zero 0. S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) -convergent ) sequence of ( ( ) ( V1() non trivial ) set ) ) ) : ( ( Function-like quasi_total ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of T : ( ( Function-like quasi_total ) ( V16() V19(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) V20(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of T : ( ( Function-like quasi_total ) ( V16() V19(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) V20(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of T : ( ( Function-like quasi_total ) ( V16() V19(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) V20(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( ) set ) ) is convergent & lim ((||.h : ( ( Function-like quasi_total non-zero 0. S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) -convergent ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total non-zero 0. S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) -convergent ) sequence of ( ( ) ( V1() non trivial ) set ) ) .|| : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ") : ( ( Function-like ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) (IT : ( ( Function-like quasi_total ) ( V16() V19(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) V20(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /* h : ( ( Function-like quasi_total non-zero 0. S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) -convergent ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total non-zero 0. S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) -convergent ) sequence of ( ( ) ( V1() non trivial ) set ) ) ) : ( ( Function-like quasi_total ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of T : ( ( Function-like quasi_total ) ( V16() V19(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) V20(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of T : ( ( Function-like quasi_total ) ( V16() V19(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) V20(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of T : ( ( Function-like quasi_total ) ( V16() V19(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) V20(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of T : ( ( Function-like quasi_total ) ( V16() V19(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) V20(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = 0. T : ( ( Function-like quasi_total ) ( V16() V19(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) V20(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of T : ( ( Function-like quasi_total ) ( V16() V19(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) V20(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) );
end;

registration
let S, T be ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ;
cluster V16() V19( the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like for ( ( ) ( ) Element of K6(K7( the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) , the carrier of T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;
end;

definition
let S, T be ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ;
mode RestFunc of S,T is ( ( Function-like RestFunc-like ) ( V16() V19( the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) PartFunc of ,) ;
end;

theorem :: NDIFF_1:23
for S, T being ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for R being ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) st R : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is total holds
( R : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is RestFunc-like iff for r being ( ( ) ( V11() real ext-real ) Real) st r : ( ( ) ( V11() real ext-real ) Real) > 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) holds
ex d being ( ( ) ( V11() real ext-real ) Real) st
( d : ( ( ) ( V11() real ext-real ) Real) > 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for z being ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) st z : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) <> 0. S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) & ||.z : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) < d : ( ( ) ( V11() real ext-real ) Real) holds
(||.z : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) ") : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) * ||.(R : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) /. z : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) < r : ( ( ) ( V11() real ext-real ) Real) ) ) ) ;

theorem :: NDIFF_1:24
for S, T being ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for R being ( ( Function-like RestFunc-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) )
for s being ( ( Function-like quasi_total non-zero 0. b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) -convergent ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total non-zero 0. b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) -convergent ) sequence of ( ( ) ( V1() non trivial ) set ) ) holds
( R : ( ( Function-like RestFunc-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) /* s : ( ( Function-like quasi_total non-zero 0. b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) -convergent ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total non-zero 0. b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) -convergent ) sequence of ( ( ) ( V1() non trivial ) set ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is convergent & lim (R : ( ( Function-like RestFunc-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) /* s : ( ( Function-like quasi_total non-zero 0. b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) -convergent ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total non-zero 0. b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) -convergent ) sequence of ( ( ) ( V1() non trivial ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) = 0. T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) ) ;

theorem :: NDIFF_1:25
for S, T being ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for h1, h2 being ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,)
for seq being ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() non trivial ) set ) ) st h1 : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is total & h2 : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is total holds
( (h1 : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) + h2 : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) , the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /* seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() non trivial ) set ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = (h1 : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() non trivial ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) + (h2 : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() non trivial ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & (h1 : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) - h2 : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) , the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /* seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() non trivial ) set ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = (h1 : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() non trivial ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) - (h2 : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() non trivial ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: NDIFF_1:26
for S, T being ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for h being ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,)
for seq being ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() non trivial ) set ) )
for r being ( ( ) ( V11() real ext-real ) Real) st h : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is total holds
(r : ( ( ) ( V11() real ext-real ) Real) (#) h : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) , the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /* seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() non trivial ) set ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = r : ( ( ) ( V11() real ext-real ) Real) * (h : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() non trivial ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: NDIFF_1:27
for T, S being ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) holds
( f : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) iff ( x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) in dom f : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) & ( for s1 being ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() non trivial ) set ) ) st rng s1 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) c= dom f : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) & s1 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() non trivial ) set ) ) is convergent & lim s1 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) = x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) & ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) holds s1 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() non trivial ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) <> x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) ) holds
( f : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) /* s1 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() non trivial ) set ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is convergent & f : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) = lim (f : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) /* s1 : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() non trivial ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) ) ) ) ) ;

theorem :: NDIFF_1:28
for T, S being ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for R1, R2 being ( ( Function-like RestFunc-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) holds
( R1 : ( ( Function-like RestFunc-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) + R2 : ( ( Function-like RestFunc-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) , the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is ( ( Function-like RestFunc-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) & R1 : ( ( Function-like RestFunc-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) - R2 : ( ( Function-like RestFunc-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) , the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is ( ( Function-like RestFunc-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) ;

theorem :: NDIFF_1:29
for T, S being ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for r being ( ( ) ( V11() real ext-real ) Real)
for R being ( ( Function-like RestFunc-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) holds r : ( ( ) ( V11() real ext-real ) Real) (#) R : ( ( Function-like RestFunc-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) , the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is ( ( Function-like RestFunc-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ;

definition
let S, T be ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ;
let f be ( ( Function-like ) ( V16() V19( the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ;
let x0 be ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) ;
pred f is_differentiable_in x0 means :: NDIFF_1:def 6
 ex N being ( ( ) ( ) Neighbourhood of x0 : ( ( Function-like quasi_total ) ( V16() V19(K7(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) V20(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) st
( N : ( ( ) ( ) Neighbourhood of x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) ) c= dom f : ( ( Function-like quasi_total ) ( V16() V19(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) V20(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V16() ) Element of K6( the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) & ex L being ( ( ) ( V16() Function-like ) Point of ( ( ) ( V1() ) set ) ) ex R being ( ( Function-like RestFunc-like ) ( V16() V19( the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) st
for x being ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) st x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) in N : ( ( ) ( ) Neighbourhood of x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) ) holds
(f : ( ( Function-like quasi_total ) ( V16() V19(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) V20(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) - (f : ( ( Function-like quasi_total ) ( V16() V19(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) V20(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x0 : ( ( Function-like quasi_total ) ( V16() V19(K7(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) V20(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) = (L : ( ( ) ( V16() Function-like ) Point of ( ( ) ( V1() ) set ) ) . (x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) - x0 : ( ( Function-like quasi_total ) ( V16() V19(K7(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) V20(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) + (R : ( ( Function-like RestFunc-like ) ( V16() V19( the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) /. (x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) - x0 : ( ( Function-like quasi_total ) ( V16() V19(K7(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) V20(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) );
end;

definition
let S, T be ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ;
let f be ( ( Function-like ) ( V16() V19( the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ;
let x0 be ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) ;
assume f : ( ( Function-like ) ( V16() V19( the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_in x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) ;
func diff (f,x0) -> ( ( ) ( V16() Function-like ) Point of ( ( ) ( V1() ) set ) ) means :: NDIFF_1:def 7
 ex N being ( ( ) ( ) Neighbourhood of x0 : ( ( Function-like quasi_total ) ( V16() V19(K7(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) V20(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) st
( N : ( ( ) ( ) Neighbourhood of x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) ) c= dom f : ( ( Function-like quasi_total ) ( V16() V19(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) V20(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V16() ) Element of K6( the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) & ex R being ( ( Function-like RestFunc-like ) ( V16() V19( the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) st
for x being ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) st x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) in N : ( ( ) ( ) Neighbourhood of x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) ) holds
(f : ( ( Function-like quasi_total ) ( V16() V19(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) V20(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) - (f : ( ( Function-like quasi_total ) ( V16() V19(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) V20(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x0 : ( ( Function-like quasi_total ) ( V16() V19(K7(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) V20(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) = (it : ( ( ) ( ) Element of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) . (x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) - x0 : ( ( Function-like quasi_total ) ( V16() V19(K7(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) V20(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) + (R : ( ( Function-like RestFunc-like ) ( V16() V19( the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) /. (x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) - x0 : ( ( Function-like quasi_total ) ( V16() V19(K7(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) V20(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) );
end;

definition
let X be ( ( ) ( ) set ) ;
let S, T be ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ;
let f be ( ( Function-like ) ( V16() V19( the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ;
pred f is_differentiable_on X means :: NDIFF_1:def 8
( X : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) c= dom f : ( ( Function-like quasi_total ) ( V16() V19(K7(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ,X : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) V20(X : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ,X : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V16() ) Element of K6( the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) st x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) in X : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) holds
f : ( ( Function-like quasi_total ) ( V16() V19(K7(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ,X : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) V20(X : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ,X : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) | X : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( Function-like ) ( V16() V19( the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of T : ( ( Function-like quasi_total ) ( V16() V19(K7(X : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,X : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) V20(X : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(X : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,X : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) Function-like ) Element of K6(K7( the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) , the carrier of T : ( ( Function-like quasi_total ) ( V16() V19(K7(X : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,X : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) V20(X : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(X : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,X : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_in x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) ) );
end;

theorem :: NDIFF_1:30
for X being ( ( ) ( ) set )
for S, T being ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b3 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b3 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_on X : ( ( ) ( ) set ) holds
X : ( ( ) ( ) set ) is ( ( ) ( ) Subset of ) ;

theorem :: NDIFF_1:31
for S, T being ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,)
for Z being ( ( ) ( ) Subset of ) st Z : ( ( ) ( ) Subset of ) is open holds
( f : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_on Z : ( ( ) ( ) Subset of ) iff ( Z : ( ( ) ( ) Subset of ) c= dom f : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) st x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) in Z : ( ( ) ( ) Subset of ) holds
f : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_in x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) ) ) ) ;

theorem :: NDIFF_1:32
for S, T being ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,)
for Y being ( ( ) ( ) Subset of ) st f : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_on Y : ( ( ) ( ) Subset of ) holds
Y : ( ( ) ( ) Subset of ) is open ;

definition
let S, T be ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ;
let f be ( ( Function-like ) ( V16() V19( the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ;
let X be ( ( ) ( ) set ) ;
assume f : ( ( Function-like ) ( V16() V19( the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_on X : ( ( ) ( ) set ) ;
func f `| X -> ( ( Function-like ) ( V16() V19( the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of (R_NormSpace_of_BoundedLinearOperators (S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) )) : ( ( non empty ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() ) set ) ) Function-like ) PartFunc of ,) means :: NDIFF_1:def 9
( dom it : ( ( ) ( ) Element of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) Element of K6( the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) = X : ( ( Function-like quasi_total ) ( V16() V19(K7(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) V20(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) st x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) in X : ( ( Function-like quasi_total ) ( V16() V19(K7(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) V20(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
it : ( ( ) ( ) Element of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) /. x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V16() Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) )) : ( ( non empty ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() ) set ) ) = diff (f : ( ( Function-like quasi_total ) ( V16() V19(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) V20(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) ) : ( ( ) ( V16() Function-like ) Point of ( ( ) ( V1() ) set ) ) ) );
end;

theorem :: NDIFF_1:33
for S, T being ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,)
for Z being ( ( ) ( ) Subset of ) st Z : ( ( ) ( ) Subset of ) is open & Z : ( ( ) ( ) Subset of ) c= dom f : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) & ex r being ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) st rng f : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) = {r : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) } : ( ( ) ( ) set ) holds
( f : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_on Z : ( ( ) ( ) Subset of ) & ( for x being ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) st x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) in Z : ( ( ) ( ) Subset of ) holds
(f : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) `| Z : ( ( ) ( ) Subset of ) ) : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of (R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() ) set ) ) Function-like ) PartFunc of ,) /. x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V16() Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() ) set ) ) = 0. (R_NormSpace_of_BoundedLinearOperators (S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V16() Function-like V68( R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) : ( ( non empty ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() ) set ) ) ) ) ;

registration
let S be ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ;
let h be ( ( Function-like quasi_total non-zero 0. S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) -convergent ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total non-zero 0. S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) -convergent ) sequence of ( ( ) ( V1() non trivial ) set ) ) ;
let n be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) ;
cluster h : ( ( Function-like quasi_total non-zero 0. S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V68(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) ) Element of the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) -convergent ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total non-zero 0. S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V68(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) ) Element of the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) -convergent ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ^\ n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like total ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like total ) set ) -> Function-like quasi_total non-zero 0. S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V68(S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) ) Element of the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) -convergent for ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() non trivial ) set ) ) ;
end;

theorem :: NDIFF_1:34
for S, T being ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) )
for N being ( ( ) ( ) Neighbourhood of x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) ) st f : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_in x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) & N : ( ( ) ( ) Neighbourhood of b4 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) ) c= dom f : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) holds
for h being ( ( Function-like quasi_total non-zero 0. b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) -convergent ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total non-zero 0. b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) -convergent ) sequence of ( ( ) ( V1() non trivial ) set ) )
for c being ( ( Function-like constant quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like constant total quasi_total ) sequence of ( ( ) ( V1() non trivial ) set ) ) st rng c : ( ( Function-like constant quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like constant total quasi_total ) sequence of ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( trivial ) Element of K6( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) = {x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) } : ( ( ) ( ) set ) & rng (h : ( ( Function-like quasi_total non-zero 0. b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) -convergent ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total non-zero 0. b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) -convergent ) sequence of ( ( ) ( V1() non trivial ) set ) ) + c : ( ( Function-like constant quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like constant total quasi_total ) sequence of ( ( ) ( V1() non trivial ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) c= N : ( ( ) ( ) Neighbourhood of b4 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) ) holds
( (f : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) /* (h : ( ( Function-like quasi_total non-zero 0. b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) -convergent ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total non-zero 0. b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) -convergent ) sequence of ( ( ) ( V1() non trivial ) set ) ) + c : ( ( Function-like constant quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like constant total quasi_total ) sequence of ( ( ) ( V1() non trivial ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) - (f : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) /* c : ( ( Function-like constant quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like constant total quasi_total ) sequence of ( ( ) ( V1() non trivial ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is convergent & lim ((f : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) /* (h : ( ( Function-like quasi_total non-zero 0. b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) -convergent ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total non-zero 0. b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) -convergent ) sequence of ( ( ) ( V1() non trivial ) set ) ) + c : ( ( Function-like constant quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like constant total quasi_total ) sequence of ( ( ) ( V1() non trivial ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) - (f : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) /* c : ( ( Function-like constant quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like constant total quasi_total ) sequence of ( ( ) ( V1() non trivial ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() ) Element of K6(REAL : ( ( ) ( V1() V49() V170() V171() V172() V176() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) = 0. T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) ) ;

theorem :: NDIFF_1:35
for T, S being ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for f1, f2 being ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) st f1 : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_in x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) & f2 : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_in x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) holds
( f1 : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) + f2 : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) , the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_in x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) & diff ((f1 : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) + f2 : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) , the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) ) : ( ( ) ( V16() Function-like ) Point of ( ( ) ( V1() ) set ) ) = (diff (f1 : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ,x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) )) : ( ( ) ( V16() Function-like ) Point of ( ( ) ( V1() ) set ) ) + (diff (f2 : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ,x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) )) : ( ( ) ( V16() Function-like ) Point of ( ( ) ( V1() ) set ) ) : ( ( ) ( V16() Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() ) set ) ) ) ;

theorem :: NDIFF_1:36
for T, S being ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for f1, f2 being ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) st f1 : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_in x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) & f2 : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_in x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) holds
( f1 : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) - f2 : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) , the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_in x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) & diff ((f1 : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) - f2 : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) , the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) ) : ( ( ) ( V16() Function-like ) Point of ( ( ) ( V1() ) set ) ) = (diff (f1 : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ,x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) )) : ( ( ) ( V16() Function-like ) Point of ( ( ) ( V1() ) set ) ) - (diff (f2 : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ,x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) )) : ( ( ) ( V16() Function-like ) Point of ( ( ) ( V1() ) set ) ) : ( ( ) ( V16() Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() ) set ) ) ) ;

theorem :: NDIFF_1:37
for T, S being ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for r being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) st f : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_in x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) holds
( r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) , the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_in x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) & diff ((r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) , the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) ) : ( ( ) ( V16() Function-like ) Point of ( ( ) ( V1() ) set ) ) = r : ( ( ) ( V11() real ext-real ) Real) * (diff (f : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ,x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) )) : ( ( ) ( V16() Function-like ) Point of ( ( ) ( V1() ) set ) ) : ( ( ) ( V16() Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() ) set ) ) ) ;

theorem :: NDIFF_1:38
for S being ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,)
for Z being ( ( ) ( ) Subset of ) st Z : ( ( ) ( ) Subset of ) is open & Z : ( ( ) ( ) Subset of ) c= dom f : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) | Z : ( ( ) ( ) Subset of ) : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) , the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = id Z : ( ( ) ( ) Subset of ) : ( ( total ) ( V16() V19(b3 : ( ( ) ( ) Subset of ) ) V20(b3 : ( ( ) ( ) Subset of ) ) Function-like one-to-one total quasi_total ) Element of K6(K7(b3 : ( ( ) ( ) Subset of ) ,b3 : ( ( ) ( ) Subset of ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
( f : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_on Z : ( ( ) ( ) Subset of ) & ( for x being ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) st x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) in Z : ( ( ) ( ) Subset of ) holds
(f : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) `| Z : ( ( ) ( ) Subset of ) ) : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of (R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() ) set ) ) Function-like ) PartFunc of ,) /. x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V16() Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() ) set ) ) = id the carrier of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) : ( ( total ) ( V1() V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like one-to-one total quasi_total ) Element of K6(K7( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) , the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: NDIFF_1:39
for S, T being ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for Z being ( ( ) ( ) Subset of ) st Z : ( ( ) ( ) Subset of ) is open holds
for f1, f2 being ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) st Z : ( ( ) ( ) Subset of ) c= dom (f1 : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) + f2 : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) , the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) & f1 : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_on Z : ( ( ) ( ) Subset of ) & f2 : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_on Z : ( ( ) ( ) Subset of ) holds
( f1 : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) + f2 : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) , the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( ) ( ) Subset of ) & ( for x being ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) st x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) in Z : ( ( ) ( ) Subset of ) holds
((f1 : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) + f2 : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) , the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( ) ( ) Subset of ) ) : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of (R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() ) set ) ) Function-like ) PartFunc of ,) /. x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V16() Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() ) set ) ) = (diff (f1 : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ,x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) )) : ( ( ) ( V16() Function-like ) Point of ( ( ) ( V1() ) set ) ) + (diff (f2 : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ,x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) )) : ( ( ) ( V16() Function-like ) Point of ( ( ) ( V1() ) set ) ) : ( ( ) ( V16() Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() ) set ) ) ) ) ;

theorem :: NDIFF_1:40
for S, T being ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for Z being ( ( ) ( ) Subset of ) st Z : ( ( ) ( ) Subset of ) is open holds
for f1, f2 being ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) st Z : ( ( ) ( ) Subset of ) c= dom (f1 : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) - f2 : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) , the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) & f1 : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_on Z : ( ( ) ( ) Subset of ) & f2 : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_on Z : ( ( ) ( ) Subset of ) holds
( f1 : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) - f2 : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) , the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( ) ( ) Subset of ) & ( for x being ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) st x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) in Z : ( ( ) ( ) Subset of ) holds
((f1 : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) - f2 : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) , the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( ) ( ) Subset of ) ) : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of (R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() ) set ) ) Function-like ) PartFunc of ,) /. x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V16() Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() ) set ) ) = (diff (f1 : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ,x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) )) : ( ( ) ( V16() Function-like ) Point of ( ( ) ( V1() ) set ) ) - (diff (f2 : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ,x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) )) : ( ( ) ( V16() Function-like ) Point of ( ( ) ( V1() ) set ) ) : ( ( ) ( V16() Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() ) set ) ) ) ) ;

theorem :: NDIFF_1:41
for S, T being ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for Z being ( ( ) ( ) Subset of ) st Z : ( ( ) ( ) Subset of ) is open holds
for r being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) st Z : ( ( ) ( ) Subset of ) c= dom (r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) , the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_on Z : ( ( ) ( ) Subset of ) holds
( r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) , the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( ) ( ) Subset of ) & ( for x being ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) st x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) in Z : ( ( ) ( ) Subset of ) holds
((r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) , the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( ) ( ) Subset of ) ) : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of (R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() ) set ) ) Function-like ) PartFunc of ,) /. x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V16() Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() ) set ) ) = r : ( ( ) ( V11() real ext-real ) Real) * (diff (f : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ,x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) )) : ( ( ) ( V16() Function-like ) Point of ( ( ) ( V1() ) set ) ) : ( ( ) ( V16() Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() ) set ) ) ) ) ;

theorem :: NDIFF_1:42
for S, T being ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,)
for Z being ( ( ) ( ) Subset of ) st Z : ( ( ) ( ) Subset of ) is open & Z : ( ( ) ( ) Subset of ) c= dom f : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) | Z : ( ( ) ( ) Subset of ) : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) , the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is constant holds
( f : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_on Z : ( ( ) ( ) Subset of ) & ( for x being ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) st x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) in Z : ( ( ) ( ) Subset of ) holds
(f : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) `| Z : ( ( ) ( ) Subset of ) ) : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of (R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() ) set ) ) Function-like ) PartFunc of ,) /. x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V16() Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() ) set ) ) = 0. (R_NormSpace_of_BoundedLinearOperators (S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V16() Function-like V68( R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) : ( ( non empty ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) ) ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() ) set ) ) ) ) ;

theorem :: NDIFF_1:43
for S being ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,)
for r being ( ( ) ( V11() real ext-real ) Real)
for p being ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) )
for Z being ( ( ) ( ) Subset of ) st Z : ( ( ) ( ) Subset of ) is open & Z : ( ( ) ( ) Subset of ) c= dom f : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) st x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) in Z : ( ( ) ( ) Subset of ) holds
f : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) /. x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) = (r : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) + p : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) ) holds
( f : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_on Z : ( ( ) ( ) Subset of ) & ( for x being ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) st x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) in Z : ( ( ) ( ) Subset of ) holds
(f : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) `| Z : ( ( ) ( ) Subset of ) ) : ( ( Function-like ) ( V16() V19( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of (R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() ) set ) ) Function-like ) PartFunc of ,) /. x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V16() Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() ) set ) ) = r : ( ( ) ( V11() real ext-real ) Real) * (FuncUnit S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) : ( ( ) ( ) Element of BoundedLinearOperators (b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) : ( ( ) ( V1() ) Element of K6( the carrier of (R_VectorSpace_of_LinearOperators (b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) )) : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V87() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RLSStruct ) : ( ( ) ( V1() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of BoundedLinearOperators (b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) : ( ( ) ( V1() ) Element of K6( the carrier of (R_VectorSpace_of_LinearOperators (b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) )) : ( ( non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty V87() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) RLSStruct ) : ( ( ) ( V1() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ;

theorem :: NDIFF_1:44
for T, S being ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) st f : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_in x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) holds
f : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) ;

theorem :: NDIFF_1:45
for X being ( ( ) ( ) set )
for S, T being ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b3 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b3 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_on X : ( ( ) ( ) set ) holds
f : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b3 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) ;

theorem :: NDIFF_1:46
for X being ( ( ) ( ) set )
for T, S being ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( V16() V19( the carrier of b3 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,)
for Z being ( ( ) ( ) Subset of ) st Z : ( ( ) ( ) Subset of ) is open & f : ( ( Function-like ) ( V16() V19( the carrier of b3 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_on X : ( ( ) ( ) set ) & Z : ( ( ) ( ) Subset of ) c= X : ( ( ) ( ) set ) holds
f : ( ( Function-like ) ( V16() V19( the carrier of b3 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_on Z : ( ( ) ( ) Subset of ) ;

theorem :: NDIFF_1:47
for T, S being ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,)
for x0 being ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) st f : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_in x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) holds
ex N being ( ( ) ( ) Neighbourhood of x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) ) st
( N : ( ( ) ( ) Neighbourhood of b4 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) ) c= dom f : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) set ) ) & ex R being ( ( Function-like RestFunc-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) st
( R : ( ( Function-like RestFunc-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) /. (0. S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) : ( ( ) ( V68(b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) = 0. T : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) & R : ( ( Function-like RestFunc-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) is_continuous_in 0. S : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V68(b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) & ( for x being ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) st x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) in N : ( ( ) ( ) Neighbourhood of b4 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) ) holds
(f : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) /. x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) - (f : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) /. x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) = ((diff (f : ( ( Function-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ,x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) )) : ( ( ) ( V16() Function-like ) Point of ( ( ) ( V1() ) set ) ) . (x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) - x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) + (R : ( ( Function-like RestFunc-like ) ( V16() V19( the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) V20( the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ,b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) ) /. (x : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) - x0 : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) ( non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) ) ) ) ;