PDIFF_2

:: PDIFF_2 semantic presentation

begin

theorem :: PDIFF_2:1

( dom (proj (1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( V1() V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional FinSequence-membered ) Element of K6((REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) = REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) & rng (proj (1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( V1() V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V160() V161() V162() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) = REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) & ( for x, y being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) holds (proj (1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( V1() V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . <*x : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ,y : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) *> : ( ( ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) set ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) = x : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ) ;

theorem :: PDIFF_2:2

( dom (proj (2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( V1() V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional FinSequence-membered ) Element of K6((REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) = REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) & rng (proj (2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( V1() V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V160() V161() V162() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) = REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) & ( for x, y being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) holds (proj (2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( V1() V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . <*x : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ,y : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) *> : ( ( ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) set ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) = y : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ) ;

begin

definition

let n, i be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( V16() V19( REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ;
let z be ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ;

func SVF1 (i,f,z) -> ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) equals :: PDIFF_2:def 1
f : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) * (reproj (i : ( ( V1() ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,z : ( ( Function-like ) ( V16() V19( the U1 of K418(n : ( ( V1() ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V66() V129() ) ( V66() V129() ) L15()) : ( ( ) ( ) set ) ) V20( the U1 of K418(i : ( ( V1() ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V66() V129() ) ( V66() V129() ) L15()) : ( ( ) ( ) set ) ) Function-like ) Element of K6(K7( the U1 of K418(n : ( ( V1() ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V66() V129() ) ( V66() V129() ) L15()) : ( ( ) ( ) set ) , the U1 of K418(i : ( ( V1() ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V66() V129() ) ( V66() V129() ) L15()) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( Function-like quasi_total ) ( V1() V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL n : ( ( V1() ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) Function-like total quasi_total ) Element of K6(K7(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ,(REAL n : ( ( V1() ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ;

end;

theorem :: PDIFF_2:3

for x, y being ( ( ) ( V11() real ext-real ) Real)
for z being ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) )
for f being ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) st z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) = <*x : ( ( ) ( V11() real ext-real ) Real) ,y : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) set ) & f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) holds
SVF1 (1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_differentiable_in x : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: PDIFF_2:4

for x, y being ( ( ) ( V11() real ext-real ) Real)
for z being ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) )
for f being ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) st z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) = <*x : ( ( ) ( V11() real ext-real ) Real) ,y : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) set ) & f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) holds
SVF1 (2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_differentiable_in y : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: PDIFF_2:5

for f being ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,)
for z being ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) holds
( ex x0, y0 being ( ( ) ( V11() real ext-real ) Real) st
( z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) = <*x0 : ( ( ) ( V11() real ext-real ) Real) ,y0 : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) set ) & SVF1 (1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_differentiable_in x0 : ( ( ) ( V11() real ext-real ) Real) ) iff f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: PDIFF_2:6

for f being ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,)
for z being ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) holds
( ex x0, y0 being ( ( ) ( V11() real ext-real ) Real) st
( z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) = <*x0 : ( ( ) ( V11() real ext-real ) Real) ,y0 : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) set ) & SVF1 (2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_differentiable_in y0 : ( ( ) ( V11() real ext-real ) Real) ) iff f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: PDIFF_2:7

for x0, y0 being ( ( ) ( V11() real ext-real ) Real)
for z being ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) )
for f being ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) st z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) = <*x0 : ( ( ) ( V11() real ext-real ) Real) ,y0 : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) set ) & f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) holds
ex N being ( ( ) ( V160() V161() V162() ) Neighbourhood of x0 : ( ( ) ( V11() real ext-real ) Real) ) st
( N : ( ( ) ( V160() V161() V162() ) Neighbourhood of b₁ : ( ( ) ( V11() real ext-real ) Real) ) c= dom (SVF1 (1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) )) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( V160() V161() V162() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) & ex L being ( ( Function-like linear ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued linear ) LinearFunc) ex R being ( ( Function-like RestFunc-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued RestFunc-like ) RestFunc) st
for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in N : ( ( ) ( V160() V161() V162() ) Neighbourhood of b₁ : ( ( ) ( V11() real ext-real ) Real) ) holds
((SVF1 (1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) )) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) - ((SVF1 (1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) )) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) . x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) = (L : ( ( Function-like linear ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued linear ) LinearFunc) . (x : ( ( ) ( V11() real ext-real ) Real) - x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) + (R : ( ( Function-like RestFunc-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued RestFunc-like ) RestFunc) . (x : ( ( ) ( V11() real ext-real ) Real) - x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ;

theorem :: PDIFF_2:8

for x0, y0 being ( ( ) ( V11() real ext-real ) Real)
for z being ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) )
for f being ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) st z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) = <*x0 : ( ( ) ( V11() real ext-real ) Real) ,y0 : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) set ) & f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) holds
ex N being ( ( ) ( V160() V161() V162() ) Neighbourhood of y0 : ( ( ) ( V11() real ext-real ) Real) ) st
( N : ( ( ) ( V160() V161() V162() ) Neighbourhood of b₂ : ( ( ) ( V11() real ext-real ) Real) ) c= dom (SVF1 (2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) )) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( V160() V161() V162() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) & ex L being ( ( Function-like linear ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued linear ) LinearFunc) ex R being ( ( Function-like RestFunc-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued RestFunc-like ) RestFunc) st
for y being ( ( ) ( V11() real ext-real ) Real) st y : ( ( ) ( V11() real ext-real ) Real) in N : ( ( ) ( V160() V161() V162() ) Neighbourhood of b₂ : ( ( ) ( V11() real ext-real ) Real) ) holds
((SVF1 (2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) )) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) . y : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) - ((SVF1 (2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) )) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) . y0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) = (L : ( ( Function-like linear ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued linear ) LinearFunc) . (y : ( ( ) ( V11() real ext-real ) Real) - y0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) + (R : ( ( Function-like RestFunc-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued RestFunc-like ) RestFunc) . (y : ( ( ) ( V11() real ext-real ) Real) - y0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ;

theorem :: PDIFF_2:9

for f being ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,)
for z being ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) holds
( f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) iff ex x0, y0 being ( ( ) ( V11() real ext-real ) Real) st
( z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) = <*x0 : ( ( ) ( V11() real ext-real ) Real) ,y0 : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) set ) & ex N being ( ( ) ( V160() V161() V162() ) Neighbourhood of x0 : ( ( ) ( V11() real ext-real ) Real) ) st
( N : ( ( ) ( V160() V161() V162() ) Neighbourhood of b₃ : ( ( ) ( V11() real ext-real ) Real) ) c= dom (SVF1 (1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) )) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( V160() V161() V162() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) & ex L being ( ( Function-like linear ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued linear ) LinearFunc) ex R being ( ( Function-like RestFunc-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued RestFunc-like ) RestFunc) st
for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in N : ( ( ) ( V160() V161() V162() ) Neighbourhood of b₃ : ( ( ) ( V11() real ext-real ) Real) ) holds
((SVF1 (1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) )) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) - ((SVF1 (1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) )) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) . x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) = (L : ( ( Function-like linear ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued linear ) LinearFunc) . (x : ( ( ) ( V11() real ext-real ) Real) - x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) + (R : ( ( Function-like RestFunc-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued RestFunc-like ) RestFunc) . (x : ( ( ) ( V11() real ext-real ) Real) - x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ) ) ;

theorem :: PDIFF_2:10

for f being ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,)
for z being ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) holds
( f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) iff ex x0, y0 being ( ( ) ( V11() real ext-real ) Real) st
( z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) = <*x0 : ( ( ) ( V11() real ext-real ) Real) ,y0 : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) set ) & ex N being ( ( ) ( V160() V161() V162() ) Neighbourhood of y0 : ( ( ) ( V11() real ext-real ) Real) ) st
( N : ( ( ) ( V160() V161() V162() ) Neighbourhood of b₄ : ( ( ) ( V11() real ext-real ) Real) ) c= dom (SVF1 (2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) )) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( V160() V161() V162() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) & ex L being ( ( Function-like linear ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued linear ) LinearFunc) ex R being ( ( Function-like RestFunc-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued RestFunc-like ) RestFunc) st
for y being ( ( ) ( V11() real ext-real ) Real) st y : ( ( ) ( V11() real ext-real ) Real) in N : ( ( ) ( V160() V161() V162() ) Neighbourhood of b₄ : ( ( ) ( V11() real ext-real ) Real) ) holds
((SVF1 (2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) )) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) . y : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) - ((SVF1 (2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) )) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) . y0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) = (L : ( ( Function-like linear ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued linear ) LinearFunc) . (y : ( ( ) ( V11() real ext-real ) Real) - y0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) + (R : ( ( Function-like RestFunc-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued RestFunc-like ) RestFunc) . (y : ( ( ) ( V11() real ext-real ) Real) - y0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ) ) ;

theorem :: PDIFF_2:11

for x0, y0, r being ( ( ) ( V11() real ext-real ) Real)
for z being ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) )
for f being ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) st z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) = <*x0 : ( ( ) ( V11() real ext-real ) Real) ,y0 : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) set ) & f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( r : ( ( ) ( V11() real ext-real ) Real) = partdiff (f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) iff ex x0, y0 being ( ( ) ( V11() real ext-real ) Real) st
( z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) = <*x0 : ( ( ) ( V11() real ext-real ) Real) ,y0 : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) set ) & ex N being ( ( ) ( V160() V161() V162() ) Neighbourhood of x0 : ( ( ) ( V11() real ext-real ) Real) ) st
( N : ( ( ) ( V160() V161() V162() ) Neighbourhood of b₆ : ( ( ) ( V11() real ext-real ) Real) ) c= dom (SVF1 (1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) )) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( V160() V161() V162() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) & ex L being ( ( Function-like linear ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued linear ) LinearFunc) ex R being ( ( Function-like RestFunc-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued RestFunc-like ) RestFunc) st
( r : ( ( ) ( V11() real ext-real ) Real) = L : ( ( Function-like linear ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued linear ) LinearFunc) . 1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in N : ( ( ) ( V160() V161() V162() ) Neighbourhood of b₆ : ( ( ) ( V11() real ext-real ) Real) ) holds
((SVF1 (1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) )) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) - ((SVF1 (1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) )) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) . x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) = (L : ( ( Function-like linear ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued linear ) LinearFunc) . (x : ( ( ) ( V11() real ext-real ) Real) - x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) + (R : ( ( Function-like RestFunc-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued RestFunc-like ) RestFunc) . (x : ( ( ) ( V11() real ext-real ) Real) - x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ) ) ) ) ;

theorem :: PDIFF_2:12

for x0, y0, r being ( ( ) ( V11() real ext-real ) Real)
for z being ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) )
for f being ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) st z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) = <*x0 : ( ( ) ( V11() real ext-real ) Real) ,y0 : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) set ) & f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( r : ( ( ) ( V11() real ext-real ) Real) = partdiff (f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) iff ex x0, y0 being ( ( ) ( V11() real ext-real ) Real) st
( z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) = <*x0 : ( ( ) ( V11() real ext-real ) Real) ,y0 : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) set ) & ex N being ( ( ) ( V160() V161() V162() ) Neighbourhood of y0 : ( ( ) ( V11() real ext-real ) Real) ) st
( N : ( ( ) ( V160() V161() V162() ) Neighbourhood of b₇ : ( ( ) ( V11() real ext-real ) Real) ) c= dom (SVF1 (2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) )) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( V160() V161() V162() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) & ex L being ( ( Function-like linear ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued linear ) LinearFunc) ex R being ( ( Function-like RestFunc-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued RestFunc-like ) RestFunc) st
( r : ( ( ) ( V11() real ext-real ) Real) = L : ( ( Function-like linear ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued linear ) LinearFunc) . 1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) & ( for y being ( ( ) ( V11() real ext-real ) Real) st y : ( ( ) ( V11() real ext-real ) Real) in N : ( ( ) ( V160() V161() V162() ) Neighbourhood of b₇ : ( ( ) ( V11() real ext-real ) Real) ) holds
((SVF1 (2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) )) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) . y : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) - ((SVF1 (2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) )) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) . y0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) = (L : ( ( Function-like linear ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued linear ) LinearFunc) . (y : ( ( ) ( V11() real ext-real ) Real) - y0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) + (R : ( ( Function-like RestFunc-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued RestFunc-like ) RestFunc) . (y : ( ( ) ( V11() real ext-real ) Real) - y0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ) ) ) ) ;

theorem :: PDIFF_2:13

for x0, y0 being ( ( ) ( V11() real ext-real ) Real)
for z being ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) )
for f being ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) st z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) = <*x0 : ( ( ) ( V11() real ext-real ) Real) ,y0 : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) set ) & f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) holds
partdiff (f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) = diff ((SVF1 (1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) )) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ;

theorem :: PDIFF_2:14

for x0, y0 being ( ( ) ( V11() real ext-real ) Real)
for z being ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) )
for f being ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) st z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) = <*x0 : ( ( ) ( V11() real ext-real ) Real) ,y0 : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) set ) & f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) holds
partdiff (f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) = diff ((SVF1 (2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) )) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,y0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ;

definition

let f be ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ;
let Z be ( ( ) ( ) set ) ;

pred f is_partial_differentiable`1_on Z means :: PDIFF_2:def 2
( Z : ( ( V1() ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) c= dom f : ( ( V1() ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( functional FinSequence-membered ) Element of K6((REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) & ( for z being ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) st z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) in Z : ( ( V1() ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) holds
f : ( ( V1() ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) | Z : ( ( V1() ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_partial_differentiable_in z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) );

pred f is_partial_differentiable`2_on Z means :: PDIFF_2:def 3
( Z : ( ( V1() ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) c= dom f : ( ( V1() ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( functional FinSequence-membered ) Element of K6((REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) & ( for z being ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) st z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) in Z : ( ( V1() ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) holds
f : ( ( V1() ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) | Z : ( ( V1() ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_partial_differentiable_in z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) );

end;

theorem :: PDIFF_2:15

for Z being ( ( ) ( functional FinSequence-membered ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) st f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable`1_on Z : ( ( ) ( functional FinSequence-membered ) Subset of ( ( ) ( ) set ) ) holds
( Z : ( ( ) ( functional FinSequence-membered ) Subset of ( ( ) ( ) set ) ) c= dom f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered ) Element of K6((REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) & ( for z being ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) st z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) in Z : ( ( ) ( functional FinSequence-membered ) Subset of ( ( ) ( ) set ) ) holds
f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ;

theorem :: PDIFF_2:16

for Z being ( ( ) ( functional FinSequence-membered ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) st f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable`2_on Z : ( ( ) ( functional FinSequence-membered ) Subset of ( ( ) ( ) set ) ) holds
( Z : ( ( ) ( functional FinSequence-membered ) Subset of ( ( ) ( ) set ) ) c= dom f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered ) Element of K6((REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) & ( for z being ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) st z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) in Z : ( ( ) ( functional FinSequence-membered ) Subset of ( ( ) ( ) set ) ) holds
f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ;

definition

let f be ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ;
let Z be ( ( ) ( ) set ) ;
assume f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable`1_on Z : ( ( ) ( ) set ) ;

func f `partial1| Z -> ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) means :: PDIFF_2:def 4
( dom it : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( functional FinSequence-membered ) Element of K6((REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) = Z : ( ( V1() ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for z being ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) st z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) in Z : ( ( V1() ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) holds
it : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) . z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) = partdiff (f : ( ( V1() ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) );

end;

definition

let f be ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ;
let Z be ( ( ) ( ) set ) ;
assume f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable`2_on Z : ( ( ) ( ) set ) ;

func f `partial2| Z -> ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) means :: PDIFF_2:def 5
( dom it : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( functional FinSequence-membered ) Element of K6((REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( ) set ) ) = Z : ( ( V1() ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for z being ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) st z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) in Z : ( ( V1() ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) holds
it : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) . z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) = partdiff (f : ( ( V1() ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) );

end;

begin

theorem :: PDIFF_2:17

for f being ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,)
for z0 being ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) )
for N being ( ( ) ( V160() V161() V162() ) Neighbourhood of (proj (1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( V1() V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) st f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) & N : ( ( ) ( V160() V161() V162() ) Neighbourhood of (proj (1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( V1() V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . b₂ : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) c= dom (SVF1 (1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) )) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( V160() V161() V162() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) holds
for h being ( ( non-zero Function-like quasi_total 0 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative functional FinSequence-membered V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() V16() non-zero V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative functional FinSequence-membered V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence)
for c being ( ( Function-like constant quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) 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() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like constant total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) : ( ( ) ( V160() V161() V162() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) = {((proj (1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( V1() V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) } : ( ( ) ( V1() V160() V161() V162() ) set ) & rng (h : ( ( non-zero Function-like quasi_total 0 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative functional FinSequence-membered V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() V16() non-zero V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative functional FinSequence-membered V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( Function-like constant quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like constant total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( Function-like ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V160() V161() V162() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) c= N : ( ( ) ( V160() V161() V162() ) Neighbourhood of (proj (1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( V1() V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . b₂ : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) holds
( (h : ( ( non-zero Function-like quasi_total 0 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative functional FinSequence-membered V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() V16() non-zero V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative functional FinSequence-membered V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) ") : ( ( Function-like ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) (((SVF1 (1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) )) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /* (h : ( ( non-zero Function-like quasi_total 0 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative functional FinSequence-membered V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() V16() non-zero V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative functional FinSequence-membered V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( Function-like constant quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like constant total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( Function-like ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - ((SVF1 (1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) )) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /* c : ( ( Function-like constant quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like constant total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is convergent & partdiff (f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) = lim ((h : ( ( non-zero Function-like quasi_total 0 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative functional FinSequence-membered V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() V16() non-zero V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative functional FinSequence-membered V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) ") : ( ( Function-like ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) (((SVF1 (1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) )) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /* (h : ( ( non-zero Function-like quasi_total 0 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative functional FinSequence-membered V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() V16() non-zero V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative functional FinSequence-membered V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( Function-like constant quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like constant total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( Function-like ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - ((SVF1 (1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) )) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /* c : ( ( Function-like constant quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like constant total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ;

theorem :: PDIFF_2:18

for f being ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,)
for z0 being ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) )
for N being ( ( ) ( V160() V161() V162() ) Neighbourhood of (proj (2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( V1() V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) st f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) & N : ( ( ) ( V160() V161() V162() ) Neighbourhood of (proj (2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( V1() V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . b₂ : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) c= dom (SVF1 (2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) )) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( V160() V161() V162() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) holds
for h being ( ( non-zero Function-like quasi_total 0 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative functional FinSequence-membered V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() V16() non-zero V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative functional FinSequence-membered V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence)
for c being ( ( Function-like constant quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) 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() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like constant total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) : ( ( ) ( V160() V161() V162() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) = {((proj (2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( V1() V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) } : ( ( ) ( V1() V160() V161() V162() ) set ) & rng (h : ( ( non-zero Function-like quasi_total 0 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative functional FinSequence-membered V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() V16() non-zero V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative functional FinSequence-membered V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( Function-like constant quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like constant total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( Function-like ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V160() V161() V162() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) c= N : ( ( ) ( V160() V161() V162() ) Neighbourhood of (proj (2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( V1() V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . b₂ : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) holds
( (h : ( ( non-zero Function-like quasi_total 0 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative functional FinSequence-membered V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() V16() non-zero V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative functional FinSequence-membered V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) ") : ( ( Function-like ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) (((SVF1 (2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) )) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /* (h : ( ( non-zero Function-like quasi_total 0 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative functional FinSequence-membered V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() V16() non-zero V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative functional FinSequence-membered V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( Function-like constant quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like constant total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( Function-like ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - ((SVF1 (2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) )) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /* c : ( ( Function-like constant quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like constant total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is convergent & partdiff (f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) = lim ((h : ( ( non-zero Function-like quasi_total 0 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative functional FinSequence-membered V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() V16() non-zero V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative functional FinSequence-membered V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) ") : ( ( Function-like ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) (((SVF1 (2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) )) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /* (h : ( ( non-zero Function-like quasi_total 0 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative functional FinSequence-membered V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() V16() non-zero V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative functional FinSequence-membered V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( Function-like constant quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like constant total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( Function-like ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - ((SVF1 (2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) )) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /* c : ( ( Function-like constant quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like constant total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( Function-like quasi_total ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ;

theorem :: PDIFF_2:19

for z0 being ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) )
for f1, f2 being ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) st f1 : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) & f2 : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) holds
f1 : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) (#) f2 : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_partial_differentiable_in z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: PDIFF_2:20

for z0 being ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) )
for f1, f2 being ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) st f1 : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) & f2 : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) holds
f1 : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) (#) f2 : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_partial_differentiable_in z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: PDIFF_2:21

for f being ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,)
for z0 being ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) st f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) holds
SVF1 (1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_continuous_in (proj (1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( V1() V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ;

theorem :: PDIFF_2:22

for f being ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,)
for z0 being ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) st f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ,2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) holds
SVF1 (2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,f : ( ( Function-like ) ( V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) ) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_continuous_in (proj (2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( V1() V16() V19( REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ,REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) Function-like complex-valued ext-real-valued real-valued V54() V61(2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like ) Element of REAL 2 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V54() V160() V161() V162() V166() ) set ) ) ;