:: NDIFF_3 semantic presentation

begin

theorem :: NDIFF_3:1
for G being ( ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace)
for s1 being ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence)
for seq being ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) st ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V112() ext-real non negative V152() V153() V154() V155() V156() V157() V158() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) holds ||.(seq : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V112() ext-real non negative V152() V153() V154() V155() V156() V157() V158() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) .|| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) <= s1 : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V112() ext-real non negative V152() V153() V154() V155() V156() V157() V158() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) ) & s1 : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & lim s1 : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V112() ext-real non negative V152() V153() V154() V155() V156() V157() V158() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) holds
( seq : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) is convergent & lim seq : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) = 0. G : ( ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V53(b1 : ( ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of b1 : ( ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) ) ;

theorem :: NDIFF_3:2
for G being ( ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace)
for k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V112() ext-real non negative V152() V153() V154() V155() V156() V157() V158() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) )
for s1 being ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence)
for seq being ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) holds (s1 : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ^\ k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V112() ext-real non negative V152() V153() V154() V155() V156() V157() V158() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) ) : ( ( ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) subsequence of b3 : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) (#) (seq : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) ^\ k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V112() ext-real non negative V152() V153() V154() V155() V156() V157() V158() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) ) : ( ( ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) subsequence of b4 : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) , the carrier of b1 : ( ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) = (s1 : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) , the carrier of b1 : ( ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) ^\ k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V112() ext-real non negative V152() V153() V154() V155() V156() V157() V158() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) : ( ( ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) subsequence of b3 : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) (#) b4 : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) sequence of ( ( ) ( V1() ) set ) ) : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) , the carrier of b1 : ( ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) ) ;

definition
let F be ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ;
let IT be ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ;
attr IT is RestFunc-like means :: NDIFF_3:def 1
( IT : ( ( ) ( ) Element of F : ( ( ) ( ) NORMSTR ) ) is total & ( for h being ( ( non-zero Function-like quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V112() ext-real non negative V152() V153() V154() V155() V156() V157() V158() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) -convergent ) ( V1() V13() non-zero V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V112() ext-real non negative V152() V153() V154() V155() V156() V157() V158() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) -convergent complex-valued ext-real-valued real-valued convergent ) Real_Sequence) holds
( (h : ( ( non-zero Function-like quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V112() ext-real non negative V152() V153() V154() V155() V156() V157() V158() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) -convergent ) ( V1() V13() non-zero V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V112() ext-real non negative V152() V153() V154() V155() V156() V157() V158() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) -convergent complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ") : ( ( Function-like ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ,REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() V13() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V1() ) set ) ) (#) (IT : ( ( ) ( ) Element of F : ( ( ) ( ) NORMSTR ) ) /* h : ( ( non-zero Function-like quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V112() ext-real non negative V152() V153() V154() V155() V156() V157() V158() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) -convergent ) ( V1() V13() non-zero V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V112() ext-real non negative V152() V153() V154() V155() V156() V157() V158() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) -convergent complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( Function-like quasi_total ) ( V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of F : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) , the carrier of F : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( V13() ) set ) ) : ( ( ) ( V1() ) set ) ) : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of F : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) , the carrier of F : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( V13() ) set ) ) : ( ( ) ( V1() ) set ) ) is convergent & lim ((h : ( ( non-zero Function-like quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V112() ext-real non negative V152() V153() V154() V155() V156() V157() V158() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) -convergent ) ( V1() V13() non-zero V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V112() ext-real non negative V152() V153() V154() V155() V156() V157() V158() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) -convergent complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ") : ( ( Function-like ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ,REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() V13() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V1() ) set ) ) (#) (IT : ( ( ) ( ) Element of F : ( ( ) ( ) NORMSTR ) ) /* h : ( ( non-zero Function-like quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V112() ext-real non negative V152() V153() V154() V155() V156() V157() V158() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) -convergent ) ( V1() V13() non-zero V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V112() ext-real non negative V152() V153() V154() V155() V156() V157() V158() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) -convergent complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( Function-like quasi_total ) ( V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of F : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) , the carrier of F : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( V13() ) set ) ) : ( ( ) ( V1() ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of F : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) , the carrier of F : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) : ( ( ) ( V13() ) set ) ) : ( ( ) ( V1() ) set ) ) : ( ( ) ( ) Element of the carrier of F : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) = 0. F : ( ( ) ( ) NORMSTR ) : ( ( ) ( V53(F : ( ( ) ( ) NORMSTR ) ) ) Element of the carrier of F : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) ) ) );
end;

registration
let F be ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ;
cluster V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like for ( ( ) ( ) Element of K6(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) , the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) ;
end;

definition
let F be ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ;
mode RestFunc of F is ( ( Function-like RestFunc-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) PartFunc of ,) ;
end;

definition
let F be ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ;
let IT be ( ( Function-like quasi_total ) ( V1() V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Function of REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) , the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) ;
attr IT is linear means :: NDIFF_3:def 2
 ex r being ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) st
for p being ( ( ) ( V11() real ext-real ) Real) holds IT : ( ( ) ( ) Element of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) . p : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( ) Element of the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) = p : ( ( ) ( V11() real ext-real ) Real) * r : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) ;
end;

registration
let F be ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ;
cluster V1() V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total linear for ( ( ) ( ) Element of K6(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) , the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) ;
end;

definition
let F be ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ;
mode LinearFunc of F is ( ( Function-like quasi_total linear ) ( V1() V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total linear ) Function of REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) , the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) ;
end;

theorem :: NDIFF_3:3
for F being ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace)
for L1, L2 being ( ( Function-like quasi_total linear ) ( V1() V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total linear ) LinearFunc of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ) holds
( L1 : ( ( Function-like quasi_total linear ) ( V1() V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total linear ) LinearFunc of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ) + L2 : ( ( Function-like quasi_total linear ) ( V1() V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total linear ) LinearFunc of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) is ( ( Function-like quasi_total linear ) ( V1() V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total linear ) LinearFunc of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ) & L1 : ( ( Function-like quasi_total linear ) ( V1() V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total linear ) LinearFunc of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ) - L2 : ( ( Function-like quasi_total linear ) ( V1() V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total linear ) LinearFunc of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) is ( ( Function-like quasi_total linear ) ( V1() V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total linear ) LinearFunc of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ) ) ;

theorem :: NDIFF_3:4
for F being ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace)
for r being ( ( ) ( V11() real ext-real ) Real)
for L being ( ( Function-like quasi_total linear ) ( V1() V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total linear ) LinearFunc of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ) holds r : ( ( ) ( V11() real ext-real ) Real) (#) L : ( ( Function-like quasi_total linear ) ( V1() V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total linear ) LinearFunc of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) is ( ( Function-like quasi_total linear ) ( V1() V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total linear ) LinearFunc of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ) ;

theorem :: NDIFF_3:5
for F being ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace)
for h1, h2 being ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,)
for seq being ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) st rng seq : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V1() V153() V154() V155() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) c= (dom h1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ) : ( ( ) ( V153() V154() V155() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) /\ (dom h2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ) : ( ( ) ( V153() V154() V155() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) : ( ( ) ( V153() V154() V155() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) holds
( (h1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) + h2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) /* seq : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) = (h1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) + (h2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) & (h1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) - h2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) /* seq : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) = (h1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) - (h2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) ) ;

theorem :: NDIFF_3:6
for F being ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace)
for h1, h2 being ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,)
for seq being ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) st h1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is total & h2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is total holds
( (h1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) + h2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) /* seq : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) = (h1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) + (h2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) & (h1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) - h2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) /* seq : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) = (h1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) - (h2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) /* seq : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) ) ;

theorem :: NDIFF_3:7
for F being ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace)
for R1, R2 being ( ( Function-like RestFunc-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ) holds
( R1 : ( ( Function-like RestFunc-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ) + R2 : ( ( Function-like RestFunc-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) is ( ( Function-like RestFunc-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ) & R1 : ( ( Function-like RestFunc-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ) - R2 : ( ( Function-like RestFunc-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) is ( ( Function-like RestFunc-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ) ) ;

theorem :: NDIFF_3:8
for F being ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace)
for r being ( ( ) ( V11() real ext-real ) Real)
for R being ( ( Function-like RestFunc-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ) holds r : ( ( ) ( V11() real ext-real ) Real) (#) R : ( ( Function-like RestFunc-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) is ( ( Function-like RestFunc-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ) ;

definition
let F be ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ;
let f be ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ;
let x0 be ( ( real ) ( V11() real ext-real ) number ) ;
pred f is_differentiable_in x0 means :: NDIFF_3:def 3
 ex N being ( ( ) ( V153() V154() V155() open ) Neighbourhood of x0 : ( ( Function-like quasi_total ) ( V13() V16(K7(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ) V17(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ) : ( ( ) ( V1() ) set ) ) ) st
( N : ( ( ) ( V153() V154() V155() open ) Neighbourhood of x0 : ( ( real ) ( V11() real ext-real ) number ) ) c= dom f : ( ( ) ( ) Element of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V153() V154() V155() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) & ex L being ( ( Function-like quasi_total linear ) ( V1() V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total linear ) LinearFunc of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) ex R being ( ( Function-like RestFunc-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) st
for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in N : ( ( ) ( V153() V154() V155() open ) Neighbourhood of x0 : ( ( real ) ( V11() real ext-real ) number ) ) holds
(f : ( ( ) ( ) Element of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) /. x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) - (f : ( ( ) ( ) Element of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) /. x0 : ( ( Function-like quasi_total ) ( V13() V16(K7(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ) V17(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ) : ( ( ) ( V1() ) set ) ) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) = (L : ( ( Function-like quasi_total linear ) ( V1() V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total linear ) LinearFunc of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ) . (x : ( ( ) ( V11() real ext-real ) Real) - x0 : ( ( Function-like quasi_total ) ( V13() V16(K7(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ) V17(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ) : ( ( ) ( V1() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) + (R : ( ( Function-like RestFunc-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ) /. (x : ( ( ) ( V11() real ext-real ) Real) - x0 : ( ( Function-like quasi_total ) ( V13() V16(K7(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ) V17(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ) : ( ( ) ( V1() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) );
end;

definition
let F be ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ;
let f be ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ;
let x0 be ( ( real ) ( V11() real ext-real ) number ) ;
assume f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_in x0 : ( ( real ) ( V11() real ext-real ) number ) ;
func diff (f,x0) -> ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) means :: NDIFF_3:def 4
 ex N being ( ( ) ( V153() V154() V155() open ) Neighbourhood of x0 : ( ( Function-like quasi_total ) ( V13() V16(K7(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ) V17(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ) : ( ( ) ( V1() ) set ) ) ) st
( N : ( ( ) ( V153() V154() V155() open ) Neighbourhood of x0 : ( ( real ) ( V11() real ext-real ) number ) ) c= dom f : ( ( ) ( ) Element of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V153() V154() V155() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) & ex L being ( ( Function-like quasi_total linear ) ( V1() V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total linear ) LinearFunc of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) ex R being ( ( Function-like RestFunc-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) st
( it : ( ( Function-like quasi_total ) ( V13() V16(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ) V17(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ) : ( ( ) ( V1() ) set ) ) = L : ( ( Function-like quasi_total linear ) ( V1() V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total linear ) LinearFunc of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ) . 1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real V112() ext-real positive non negative V152() V153() V154() V155() V156() V157() V158() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in N : ( ( ) ( V153() V154() V155() open ) Neighbourhood of x0 : ( ( real ) ( V11() real ext-real ) number ) ) holds
(f : ( ( ) ( ) Element of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) /. x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) - (f : ( ( ) ( ) Element of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) /. x0 : ( ( Function-like quasi_total ) ( V13() V16(K7(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ) V17(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ) : ( ( ) ( V1() ) set ) ) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) = (L : ( ( Function-like quasi_total linear ) ( V1() V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total linear ) LinearFunc of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ) . (x : ( ( ) ( V11() real ext-real ) Real) - x0 : ( ( Function-like quasi_total ) ( V13() V16(K7(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ) V17(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ) : ( ( ) ( V1() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) + (R : ( ( Function-like RestFunc-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ) /. (x : ( ( ) ( V11() real ext-real ) Real) - x0 : ( ( Function-like quasi_total ) ( V13() V16(K7(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ) V17(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ) : ( ( ) ( V1() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) ) ) );
end;

definition
let F be ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ;
let f be ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ;
let X be ( ( ) ( ) set ) ;
pred f is_differentiable_on X means :: NDIFF_3:def 5
( X : ( ( Function-like quasi_total ) ( V13() V16(K7(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ) V17(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ) : ( ( ) ( V1() ) set ) ) c= dom f : ( ( ) ( ) Element of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V153() V154() V155() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in X : ( ( Function-like quasi_total ) ( V13() V16(K7(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ) V17(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ) : ( ( ) ( V1() ) set ) ) holds
f : ( ( ) ( ) Element of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) | X : ( ( Function-like quasi_total ) ( V13() V16(K7(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ) V17(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ) : ( ( ) ( V1() ) set ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) , the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) is_differentiable_in x : ( ( ) ( V11() real ext-real ) Real) ) );
end;

theorem :: NDIFF_3:9
for F being ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_on X : ( ( ) ( ) set ) holds
X : ( ( ) ( ) set ) is ( ( ) ( V153() V154() V155() ) Subset of ( ( ) ( V1() ) set ) ) ;

theorem :: NDIFF_3:10
for F being ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace)
for Z being ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) )
for f being ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) holds
( f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) iff ( Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) c= dom f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) : ( ( ) ( V153() V154() V155() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) holds
f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_in x : ( ( ) ( V11() real ext-real ) Real) ) ) ) ;

theorem :: NDIFF_3:11
for F being ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace)
for Y being ( ( ) ( V153() V154() V155() ) Subset of ( ( ) ( V1() ) set ) )
for f being ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_on Y : ( ( ) ( V153() V154() V155() ) Subset of ( ( ) ( V1() ) set ) ) holds
Y : ( ( ) ( V153() V154() V155() ) Subset of ( ( ) ( V1() ) set ) ) is open ;

definition
let F be ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ;
let f be ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ;
let X be ( ( ) ( ) set ) ;
assume f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_on X : ( ( ) ( ) set ) ;
func f `| X -> ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) means :: NDIFF_3:def 6
( dom it : ( ( Function-like quasi_total ) ( V13() V16(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ) V17(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ) : ( ( ) ( V1() ) set ) ) : ( ( ) ( V13() V153() V154() V155() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) = X : ( ( Function-like quasi_total ) ( V13() V16(K7(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ) V17(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ) : ( ( ) ( V1() ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in X : ( ( Function-like quasi_total ) ( V13() V16(K7(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ) V17(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ) : ( ( ) ( V1() ) set ) ) holds
it : ( ( Function-like quasi_total ) ( V13() V16(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ) V17(F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ,F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V13() ) set ) ) : ( ( ) ( V1() ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( ) set ) = diff (f : ( ( ) ( ) Element of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) ,x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) ) );
end;

theorem :: NDIFF_3:12
for F being ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace)
for Z being ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) )
for f being ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) st Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) c= dom f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) : ( ( ) ( V153() V154() V155() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) & ex r being ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) st rng f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() ) set ) ) = {r : ( ( ) ( V11() real ext-real ) Real) } : ( ( ) ( V153() V154() V155() ) set ) holds
( f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) holds
(f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) `| Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) /. x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) = 0. F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V53(b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) ) ) ;

theorem :: NDIFF_3:13
for F being ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,)
for x0 being ( ( real ) ( V11() real ext-real ) number )
for N being ( ( ) ( V153() V154() V155() open ) Neighbourhood of x0 : ( ( real ) ( V11() real ext-real ) number ) ) st f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_in x0 : ( ( real ) ( V11() real ext-real ) number ) & N : ( ( ) ( V153() V154() V155() open ) Neighbourhood of b3 : ( ( real ) ( V11() real ext-real ) number ) ) c= dom f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) : ( ( ) ( V153() V154() V155() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) holds
for h being ( ( non-zero Function-like quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V112() ext-real non negative V152() V153() V154() V155() V156() V157() V158() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) -convergent ) ( V1() V13() non-zero V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V112() ext-real non negative V152() V153() V154() V155() V156() V157() V158() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) -convergent complex-valued ext-real-valued real-valued convergent ) Real_Sequence)
for c being ( ( Function-like constant quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like constant total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) st rng c : ( ( Function-like constant quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like constant total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) : ( ( ) ( V1() V153() V154() V155() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) = {x0 : ( ( real ) ( V11() real ext-real ) number ) } : ( ( ) ( V153() V154() V155() ) set ) & rng (h : ( ( non-zero Function-like quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V112() ext-real non negative V152() V153() V154() V155() V156() V157() V158() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) -convergent ) ( V1() V13() non-zero V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V112() ext-real non negative V152() V153() V154() V155() V156() V157() V158() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) -convergent complex-valued ext-real-valued real-valued convergent ) Real_Sequence) + c : ( ( Function-like constant quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like constant total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( Function-like ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ,REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() V13() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V1() ) set ) ) : ( ( ) ( V1() V153() V154() V155() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) c= N : ( ( ) ( V153() V154() V155() open ) Neighbourhood of b3 : ( ( real ) ( V11() real ext-real ) number ) ) holds
( (h : ( ( non-zero Function-like quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V112() ext-real non negative V152() V153() V154() V155() V156() V157() V158() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) -convergent ) ( V1() V13() non-zero V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V112() ext-real non negative V152() V153() V154() V155() V156() V157() V158() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) -convergent complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ") : ( ( Function-like ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ,REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() V13() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V1() ) set ) ) (#) ((f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) /* (h : ( ( non-zero Function-like quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V112() ext-real non negative V152() V153() V154() V155() V156() V157() V158() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) -convergent ) ( V1() V13() non-zero V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V112() ext-real non negative V152() V153() V154() V155() V156() V157() V158() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) -convergent complex-valued ext-real-valued real-valued convergent ) Real_Sequence) + c : ( ( Function-like constant quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like constant total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( Function-like ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ,REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() V13() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V1() ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) - (f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) /* c : ( ( Function-like constant quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like constant total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) is convergent & diff (f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ,x0 : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) = lim ((h : ( ( non-zero Function-like quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V112() ext-real non negative V152() V153() V154() V155() V156() V157() V158() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) -convergent ) ( V1() V13() non-zero V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V112() ext-real non negative V152() V153() V154() V155() V156() V157() V158() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) -convergent complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ") : ( ( Function-like ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ,REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() V13() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V1() ) set ) ) (#) ((f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) /* (h : ( ( non-zero Function-like quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V112() ext-real non negative V152() V153() V154() V155() V156() V157() V158() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) -convergent ) ( V1() V13() non-zero V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V112() ext-real non negative V152() V153() V154() V155() V156() V157() V158() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) -convergent complex-valued ext-real-valued real-valued convergent ) Real_Sequence) + c : ( ( Function-like constant quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like constant total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( Function-like ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ,REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() V13() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V1() ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) - (f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) /* c : ( ( Function-like constant quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) Function-like constant total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V13() V16( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) ) ;

theorem :: NDIFF_3:14
for F being ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace)
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f1, f2 being ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) st f1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_in x0 : ( ( ) ( V11() real ext-real ) Real) & f2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_in x0 : ( ( ) ( V11() real ext-real ) Real) holds
( f1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) + f2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) is_differentiable_in x0 : ( ( ) ( V11() real ext-real ) Real) & diff ((f1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) + f2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) = (diff (f1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) + (diff (f2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) ) ;

theorem :: NDIFF_3:15
for F being ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace)
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f1, f2 being ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) st f1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_in x0 : ( ( ) ( V11() real ext-real ) Real) & f2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_in x0 : ( ( ) ( V11() real ext-real ) Real) holds
( f1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) - f2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) is_differentiable_in x0 : ( ( ) ( V11() real ext-real ) Real) & diff ((f1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) - f2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) = (diff (f1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) - (diff (f2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) ) ;

theorem :: NDIFF_3:16
for F being ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace)
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,)
for r being ( ( ) ( V11() real ext-real ) Real) st f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_in x0 : ( ( ) ( V11() real ext-real ) Real) holds
( r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) is_differentiable_in x0 : ( ( ) ( V11() real ext-real ) Real) & diff ((r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) = r : ( ( ) ( V11() real ext-real ) Real) * (diff (f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) ) ;

theorem :: NDIFF_3:17
for F being ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace)
for Z being ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) )
for f1, f2 being ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) st Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) c= dom (f1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) + f2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) : ( ( ) ( V153() V154() V155() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) & f1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) & f2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) holds
( f1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) + f2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) is_differentiable_on Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) holds
((f1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) + f2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) `| Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( ) set ) = (diff (f1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ,x : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) + (diff (f2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ,x : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) ) ) ;

theorem :: NDIFF_3:18
for F being ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace)
for Z being ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) )
for f1, f2 being ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) st Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) c= dom (f1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) - f2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) : ( ( ) ( V153() V154() V155() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) & f1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) & f2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) holds
( f1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) - f2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) is_differentiable_on Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) holds
((f1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) - f2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) `| Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( ) set ) = (diff (f1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ,x : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) - (diff (f2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ,x : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) ) ) ;

theorem :: NDIFF_3:19
for F being ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace)
for r being ( ( ) ( V11() real ext-real ) Real)
for Z being ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) )
for f being ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) st Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) c= dom (r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) : ( ( ) ( V153() V154() V155() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) & f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) holds
( r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) is_differentiable_on Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) holds
((r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) `| Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( ) set ) = r : ( ( ) ( V11() real ext-real ) Real) * (diff (f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ,x : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) ) ) ;

theorem :: NDIFF_3:20
for F being ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace)
for Z being ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) )
for f being ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) st Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) c= dom f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) : ( ( ) ( V153() V154() V155() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) & f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) | Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) : ( ( Function-like ) ( V13() V16(b2 : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) ) V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) is constant holds
( f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) holds
(f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) `| Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( ) set ) = 0. F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V53(b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) ) ) ;

theorem :: NDIFF_3:21
for F being ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace)
for r, p being ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) )
for Z being ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) )
for f being ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) st Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) c= dom f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) : ( ( ) ( V153() V154() V155() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) holds
f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) /. x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) = (x : ( ( ) ( V11() real ext-real ) Real) * r : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) + p : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) ) holds
( f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) holds
(f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) `| Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( ) set ) = r : ( ( ) ( ) Point of ( ( ) ( V1() non trivial ) set ) ) ) ) ;

theorem :: NDIFF_3:22
for F being ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,)
for x0 being ( ( ) ( V11() real ext-real ) Real) st f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_in x0 : ( ( ) ( V11() real ext-real ) Real) holds
f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: NDIFF_3:23
for F being ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_on X : ( ( ) ( ) set ) holds
f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) | X : ( ( ) ( ) set ) : ( ( Function-like ) ( V13() V16(b2 : ( ( ) ( ) set ) ) V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) Element of K6(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) , the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) is continuous ;

theorem :: NDIFF_3:24
for F being ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace)
for X being ( ( ) ( ) set )
for Z being ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) )
for f being ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_on X : ( ( ) ( ) set ) & Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) c= X : ( ( ) ( ) set ) holds
f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) ;

theorem :: NDIFF_3:25
for F being ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ex R being ( ( Function-like RestFunc-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ) st
( R : ( ( Function-like RestFunc-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ) /. 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V112() ext-real non negative V152() V153() V154() V155() V156() V157() V158() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) = 0. F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V53(b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) & R : ( ( Function-like RestFunc-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like RestFunc-like ) RestFunc of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ) is_continuous_in 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V112() ext-real non negative V152() V153() V154() V155() V156() V157() V158() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V153() V154() V155() V156() V157() V158() V159() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ) ) ;

definition
let F be ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ;
let f be ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like ) PartFunc of ,) ;
attr f is differentiable means :: NDIFF_3:def 7
f : ( ( ) ( ) Element of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) is_differentiable_on dom f : ( ( ) ( ) Element of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( V153() V154() V155() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) ;
end;

registration
let F be ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) ;
cluster V1() V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) Function-like total quasi_total differentiable for ( ( ) ( ) Element of K6(K7(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) , the carrier of F : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V1() non trivial ) set ) ) : ( ( ) ( V1() V13() ) set ) ) : ( ( ) ( V1() ) set ) ) ;
end;

theorem :: NDIFF_3:26
for F being ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace)
for Z being ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) )
for f being ( ( Function-like differentiable ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like differentiable ) PartFunc of ,) st Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) c= dom f : ( ( Function-like differentiable ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like differentiable ) PartFunc of ,) : ( ( ) ( V153() V154() V155() ) Element of K6(REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) : ( ( ) ( V1() ) set ) ) holds
f : ( ( Function-like differentiable ) ( V13() V16( REAL : ( ( ) ( V1() V34() V153() V154() V155() V159() ) set ) ) V17( the carrier of b1 : ( ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) ( non empty non trivial V72() V97() V98() V99() V100() V101() V102() V103() V107() V108() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V1() non trivial ) set ) ) Function-like differentiable ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( V153() V154() V155() open ) Subset of ( ( ) ( V1() ) set ) ) ;